We study lattice trees (LTs) and animals (LAs) on the nearest-neighbor lattice $\Zd$  in high dimensions. We prove that LTs and LAs display mean-field behavior above dimension $\dmintree$ and $\dminanimal$, respectively. Such results have previously been obtain by Hara and Slade in sufficiently high dimensions. The dimension above which their results apply was not yet specified. We rely on the non-backtracking lace expansion (NoBLE) method that we have recently developed. The NoBLE makes use of an alternative lace expansion for LAs and LTs that perturbs around non-backtracking random walk rather than simple random walk, leading to smaller corrections. The NoBLE method then provides a careful computational analysis that improves the dimension above which the result applies. Universality arguments predict that the upper critical dimension, above which our results apply, is equal to $d_c=8$ for both models, as is known for sufficiently spread-out models by the results of Hara and Slade mentioned earlier.

The main ingredients in this paper are  (a) to derive a non-backtracking lace expansion for the LT and LA two-point functions;  (b) to bound the non-backtracking lace-expansion coefficients, thus showing that our general NoBLE methodology can be applied; and (c) to obtain sharp numerical bounds on the coefficients. Our proof is complemented by a computer-assisted numerical analysis that verifies that the necessary bounds used in the NoBLE are satisfied.
This extended version includes the definition of all bounding diagrams, see Appendix C.

§ INTRODUCTION

§.§ Motivation

Lattice trees (LTs) and lattice animals (LAs) are models for branched polymers, where excluded volume creates a self-avoidance interaction between different pieces (vertices or bonds), while the branching nature corresponds to polymers whose building blocks can have covalent bonds to more than two other building blocks. They are the branching equivalents of self-avoiding walks (see [33] for a detailed introduction, and a brief introduction to LTs and LAs as well). The study of LTs and LAs in low dimensions is quite hard and few results exist. Exceptions are the beautiful relation between  LTs in dimension $d$ and a hard-core lattice gas in dimension $d+2$, as predicted by Parisi and Sourlas [37] and shown for a continuum model by Brydges and Imbrie [5] (see also [31] for a simple and elegant proof). As a result, also LTs and LAs in other dimensions have been investigated.

Like many statistical physics models, LTs and LAs are expected to have a so-called upper critical dimension, above which their behavior should be similar to a simpler model having less intricate interactions. For LTs and LAs, the upper critical dimension is believed to be $d_c=8$, while this simpler model is branching random walk (BRW). BRW has gained enormous popularity in the probability community, and its critical behavior can be proved rigorously. See for example Perkins [38] for super-process limits of BRWs. The main tool to prove that LTs and LAs show mean-field behavior is the so-called lace expansion, a perturbation expansion that compares LTs and LAs to BRWs. Many results have so far been proved for LTs, including the identification of some of their critical exponents by Hara and Slade [20] (which also applies to LAs), scaling limits of lattice trees of a given number of vertices by Derbez and Slade [7, 8], and the structure of lattice trees that are conditioned to have long paths in them by Holmes [27] (see also [24] for a recent tightness result). Almost all of these results apply to the technically simpler setting of spread-out LTs and LAs. In this paper, our goal is to study nearest-neighbor LTs and LAs above the upper critical dimension $d_c=8$. Nearest-neighbor models are the simplest models to formulate, which makes them quite popular.

The lace expansion was first used by Brydges and Spencer in 1985 [6] to prove mean-field behavior for weakly self-avoiding walk. Since then, it has been applied (strictly) self-avoiding walks (SAW), percolation, and lattice trees and animals [17, 18, 21, 41]. It has become one of the key tools (and often the only tool available) to prove mean-field behavior of statistical mechanical models above the upper critical dimension of these models. More recent extensions include oriented percolation <cit.>, the contact process [25, 39], and the Ising model [40].

The lace expansion is a perturbative method in nature, and therefore, applications of the lace expansion typically necessitate a small parameter. This small parameter tends to be the degree of the underlying base graph. This is the reason why it is often convenient to work with so-called spread-out models, where long- but finite-range connections over a (large) distance $L$ are possible, as then the degree of the base graph can be made large independently of the dimension. Thus, results can often be proved to hold, for $L$ sufficiently large, all the way up to the critical dimension of the corresponding model. The simplest, and most often studied, version of these models is, however, the nearest-neighbor model. For the nearest-neighbor model, the degree of a vertex is $2d$. Taking $2d$ large in order to obtain a small perturbation parameter then necessitates to take $d$ large in order to prove mean-field results. This leads to suboptimal results in terms of the dimension above which the results hold. A seminal exception is SAW, where Hara and Slade [19] have proved that $d\geq 5$ is sufficient for their perturbation analysis to hold, using a computer-assisted method and a detailed analysis of the perturbation terms arising through the lace expansion. For SAW, mean-field results are expected to be false in dimension $d=4$. See the work using the renormalization group to identify the logarithmic corrections to mean-field behavior by Bauerschmidt, Brydges and Slade in [1, 2] and the references therein.

In this paper, we apply lace-expansion methodology to nearest-neighbor LTs and LAs. Hara and Slade [18] derived a lace expansion for LTs and LAs, and used it to prove mean-field behavior for sufficiently spread-out LTs and LAs in the optimal $d>8$, while their results also apply to the nearest-neighbor setting in sufficiently high dimensions. Which dimension suffices was not answered by Hara and Slade, and we revisit this question.  We derive a so-called non-backtracking lace expansion (NoBLE) and build upon the techniques of [21] to prove mean-field behavior for LTs in $d\geq \dmintree$ and for LAs for $d\geq \dminanimal$. This extends our recent work deriving general conditions under which such a NoBLE can be applied [12], and its application to percolation for $d>10$  [13].

The main differences of the NoBLE to the classical lace expansion method are that (i) we perturb around non-backtracking random walk, rather than around simple random walk, so that the lace-expansion coefficients are significantly smaller than in the classical lace expansion as used by Hara and Slade; (ii) our bounds on the lace-expansion coefficients are matrix-based, so as to profit maximally from the fact that loops present in the lace-expansion coefficients consist of at least four bonds in the NoBLE; and (iii) we use and provide Mathematica notebooks that implement the bounds, and that can be downloaded from the first author's website. We prove that the LT and LA two-point functions satisfy an infra-red bound that describes its singularity for small Fourier variables. Such an infrared bound immediately implies the finiteness of the so-called square diagram, and thus implies the existence of certain critical exponents, that take on mean-field values.

Our proof is computer-assisted, and relies on two key ingredients, that were also used to analyze percolation in high-dimensions [13]:

(I) Rigorous upper bounds on various simple random walk integrals, as first proved by Hara and Slade in [21]. This part of the analysis is unchanged compared to the Hara-Slade proof for SAWs. The crucial reason why we can use these integrals is that the non-backtracking random walk Green's function can be explicitly described in terms of the simple random walk Green's function. Our analysis requires us to compute $140$ such integrals, corresponding to convolutions of random walk Green's functions with itself at various values in $\Z^d$. We further need to compute the number of simple random walks of lengths up to $10$ ending at various points in $\Z^d$, as well as the number of related self-avoiding walks and bond-self-avoiding walks.  These bounds are performed in one Mathematica notebook;

(II) Two additional Mathematica notebooks. The first implements the computations in our general approach to the non-backtracking lace expansion (NoBLE) in [12], and the second computes the rigorous bounds on the lace-expansion coefficients provided in the present paper, both for LTs as well as for LAs. These notebooks do nothing else than implement the bounds proved here and in [12], and rely on nothing but (many) multiplications, additions, as well as diagonalisations of two five-by-five matrices. These computations could be performed by hand, but the use of the notebooks tremendously simplifies them.

Let us remark that in dimensions $d\geq 100$, say, we require only the values of $5$ integrals, and no computations concerning the number of random walks ending at various locations. The additional computations are used to obtain a sharper bound on the perturbations. We next introduce the nearest-neighbor LTs and LAs that we investigate, and state our main results. Our methods initially apply to any specific dimension, but we also derive bounds that are monotone in the dimension and can be applies to all $d\geq 30$ at once. Thus, we can check the necessary dimensions up to $d=29$ one by one, and then analyse $d\geq 30$ in one go.

§.§ The model

A nearest-neighbor lattice tree on $\Zd$ is a finite connected set of nearest-neighbor bonds that contains no cycles (closed loops).
A nearest-neighbor lattice animal on $\Zd$ is a finite connected set of nearest-neighbor bonds, which may or may not contain cycles.
Although a tree/animal $A$ is defined as a set of bonds, for $x\in\Zd$, we write $x\in A$ to denote that $x$ is an element of a bond of $A$. The number of bonds in $A$ is denoted by $|A|$. We define $t^{\sss(a)}_n(x)$ and $t^{\sss(t)}_n(x)$ to be the number of LAs and LTs, respectively, that consist of exactly $n$ bonds and contain the origin and $x\in\Zd$.
We study LAs and LTs using the one-point function $g_z$ and the two-point function $\bar G_z$, that are defined by

\begin{align}
\lbeq{defLTLAOnePoint}
&g^{\sss(a)}_z=\bar G^{\sss(a)}_z(0)=\sum_{A\colon A\ni 0}z^{|A|}, &g^{\sss(t)}_z={\bar G}^{\sss(t)}_z(0)=\sum_{T\colon 0\in T}z^{|T|},\\
\lbeq{defLTLATwoPoint}
&\bar G^{\sss(a)}_z(x)=\sum_{n=0}^\infty t^{\sss(a)}_n(x) z^n=\sum_{A\colon 0,x\in A}z^{|A|},&\bar G^{\sss(t)}_z(x)=\sum_{n=0}^\infty t^{\sss(t)}_n(x) z^n=\sum_{x\in\Zd}\sum_{T\colon 0,x\in T}z^{|T|},
\end{align}

where we sum over animals $A$ and trees $T$ respectively, and we choose $z$ such that the above sums make sense. In what follows, we drop the superscripts $(a)$ and $(t)$ when we speak about both models simultaneously. Only when we discuss specific statements for LAs or LTs separately, the superscripts will be shown. We define the susceptibility of LAs and LTs by

\begin{align}
\lbeq{defLTLAsusceptibility}
\chi(z)=\sum_{x}\sum_{A\colon 0,x\in A} z^{|A|}
\end{align}

and denote the radii of convergence of these sums by $z_c=z_c(d)$. As for SAW, $1/z_c$ describes the exponential growth of the number of LTs/LAs as $n$ grows.

We use the notation $\bar G_z$ for the two-point function, and use a normalized two-point function $G_z(x)=\bar G_z(x)/g_z$ for our analysis. We give the reason for this in Section <ref>. Intuitively, this is due to the fact that it is convenient to have that $G_z(0)=1,$ which is true for the normal two-point functions of SAW and percolation, but not for those of LTs and LAs. Since $\bar G_z(0)=g_z$ by definition, we thus prefer to work with $G_z(x)=\bar G_z(x)/g_z$ instead. Dealing properly with such one-point functions is a major ingredient of our proof.

Fourier transforms.
Our analysis makes heavy use of Fourier analysis. Unless specified otherwise, $k$ always denotes an arbitrary element from the Fourier dual of the discrete lattice, which is the torus $[-\pi,\pi]^d$. The Fourier transform of a function $f\colon\Z^d\to\C$ is defined by

f̂(k)=∑_x∈f(x) ^k·x.

For two summable function $f,g\colon \Zd \mapsto \Rbold$, we let $f\star g$ denote their convolution, i.e.,

(f⋆g)(x)= ∑_x∈ f(y)g(x-y).

We note that the Fourier transform of $f\star g$ is given by the product of $\hat f$ and $\hat g$. In particular, let $D(x)=\indic{|x|=1}/(2d)$ be the nearest-neighbor random walk transition probability, so that

D̂(k)=1/2d∑_x|x|=1 ^k·x = 1/d ∑_i=1^d cos(k_i).

Critical exponents.
It is believed that the asymptotic behavior of lattice trees and lattice animals can be described by a small number of critical exponents.
These critical exponents describe the growth of $t_n$ as $n\to\infty$ and $\bar G_z$ as $z$ approaches its radius of convergence $z_c$.
Further, they describe the typical length-scale of a lattice tree/animal, as characterized by the average radius of gyration $R_n$
and the correlation length of order two $\xi_2$, that are defined by

R_n^2= 1/2 ∑_x∈ t_n(x) ∑_x∈ |x|^2 t_n(x)
= 1/2 t̂_n(0) ∑_x∈ |x|^2 t_n(x),

=∑_n≥0 R_nt̂_n(0) z^n/∑_n≥0t̂_n(0) z^n,

where $|x|=(\sum_{i=1}^d x_i^2)^{1/2}$ denotes the Euclidean norm of $x\in \Zd$.

It is believed that there exist $\gamma,\eta,\nu$ such that

\begin{align}
\lbeq{genTreestatement1}
\chi(z)&\sim \frac {1} {(1-z/z_c)^\gamma},
\qquad
\text{ and } \qquad \xi_2(z)\sim \frac {1} {(1-z/z_c)^\nu} \text{ as } z\nearrow z_c,\\
\lbeq{genTreestatement2}
\bar G_{z_c}(x)&\sim \frac {1} {|x|^{2-d-\eta}},\qquad\text{ and }\qquad \hat {\bar G}_{z_c}(k)\sim |k|^{\eta-2} ,
\end{align}

as $|x|\to\infty$ and $k\to 0$. These exponents are believed to be universal, in the sense that they do not depend on the detailed structure of the lattice. In particular, it is believed that the values of $\gamma,\eta$ and $\nu$ are the same in the nearest-neighbor setting, that we consider here, and in the spread-out setting.
The symbol $\sim$ in genTreestatement1-genTreestatement2 can have several meanings,
and we shall always mean that the critical exponent exists in the bounded-ratio sense, meaning that there exist $0<c_1<c_2<\infty$ such that, uniformly for $z\in(0,z_c)$,

c_1/(1-z/z_c)^γ≤χ(z) ≤c_2/(1-z/z_c)^γ.

As for SAW it is believed that the critical exponents $\gamma, \nu, \eta$ are related by the Fisher relation $\gamma=(2-\eta)\nu$ and it has been proven in [4] that $\gamma\geq 1/2$ in all dimensions. Further, it is believed that there exists an upper critical dimension $d_c$ such that the critical exponents of LT/LA in $d>d_c$ take their mean-field values, which are $\gamma=1/2, \nu=1/4, \eta=0$. These values correspond to the mean-field model of LTs and LAs, studied in [3].
It is conjectured in [32] that the upper critical dimensions of LTs and LAs are $d_c=8$. In [29], site LAs and LTs are simulated and the conjectured values of the critical exponents consistent with these simulations (see [29] or [30] for the precise definition of site LAs and LTs). These computations are only done for site LTs, but as the critical exponents are expected to be universal, the values should be the same as for the bond tree/animal discussed here.
This conjecture is supported by rigorous work in [22], where it is shown that if the “square diagram” is finite at the critical point, as is believed for $d>8$, then the critical exponent $\gamma$ is at most $1/2$. Hara and Slade [18] proved that mean-field behavior holds for LTs and LAs in the spread-out setting with $L$ sufficiently large and $d>8$, or for nearest-neighbor LTs and LAs in sufficiently high dimensions. The main aim of this paper is to put an exact value to what `sufficiently large' means. The authors have learnt through private communication with Takashi Hara that Hara and Slade expected the classical lace expansion to only be successful in dimensions much larger than $d_c=8$.

§.§ Results

The main result of this paper is the following infrared bound:

For nearest-neighbor lattice trees and lattice animals in dimensions $d\geq \dmintree$ and $d\geq \dminanimal$, respectively, there exist constants $\bar{A}_1(d)$ and $\bar{A}_2(d)$ such that

uniformly for $z\leq z_c(d)$.

Note that Theorem <ref> implies that $\eta=0$ in Fourier-space bound genTreestatement2, as $[1-\hat{D}(k)]\approx |k|^2$ for small $k$.
We prove Theorem <ref> first for $\hat{G}_z(k)$ as this is the central quantity in the NoBLE analysis.
The analysis used for the proof  for $\hat{G}_z(k)$  creates bounds on the amplitudes $A_1(d)$ and $A_2(d)$ replacing $\bar{A}_1(d)$ and $\bar{A}_2(d)$. Since $\bar{G}_z(x)=g_z G_z(x)$, these bounds, together with an upper bound on $g_z$, imply upper bounds on $\hat{\bar{G}}_z(x)$ ,$\bar{A}_1(d)$ and $\bar{A}_2(d)$.

Our methods require a detailed analysis of both the critical value as well as the amplitudes $A_1(d)$ and $A_2(d)$. As a result, we obtain the following numerical bounds:

For nearest-neighbor lattice trees the following upper bounds hold:

$d$                 16    17       18     19      20      30

$\max\{\bar{A}_1(d),\bar{A}_2(d)\}\leq$         3.872    3.501     3.37       3.284     3.222     2.973

$\frac 1 {g_{z_c}} \max\{\bar{A}_1(d),\bar{A}_2(d)\}\leq$         1.293   1.182     1.147       1.125    1.11    1.052

$g_{z_c}\leq$         2.9963     2.9619     2.9383     2.9196      2.9043     2.8268

$(2d-1)g_{z_c} z_c(d)\leq$          1.1023    1.0897    1.081     1.0741     1.0685      1.04

For nearest-neighbor lattice animals the following upper bounds hold:

$d$                  17     18     19      20      30

$\max\{\bar{A}_1(d),\bar{A}_2(d)\}\leq$          3.587    3.41     3.309     3.24      2.975

$\frac 1 {g_{z_c}} \max\{\bar{A}_1(d),\bar{A}_2(d)\}\leq$          1.21     1.158      1.132    1.114    1.053

$g_{z_c}\leq$           2.9721   2.9454     2.925      2.9086    2.8277

$(2d-1)g_{z_c} z_c(d)\leq$          1.0934   1.0836    1.0761    1.07     1.0403

These upper bounds are a by-product of our analysis and we explain in Section <ref> how they are computed using the Mathematica notebooks available at [9].
We are able to improve the numerical bounds in Theorem <ref>, with reasonable efforts, but yielding an insignificant improvement. These efforts, however, would not reduce the minimal dimensions above which our results apply. It would, probably, be possible to improve these optimal dimensions slightly, but the effort required for this would be enormous. In particular, we do not expect that our results can be improved to all $d>d_c=8$ without using substantially new ingredients and insights that are way beyond our proof.

□(z_c)   =(G̅_z_c⋆G̅_z_c⋆G̅_z_c⋆G̅_z_c)(0)
=lim_z↗z_c (G̅_z⋆G̅_z⋆G̅_z⋆G̅_z)(0)

=lim_z↗z_c ∫_(-π,π)^d dk/(2π)^d Ĝ̅̂_z(k)^4,
the infrared bound in Theorem <ref> (which is uniform in $z<z_c$) immediately implies that the square condition holds, and by [22] thus also that $\gamma=1/2$:

For nearest-neighbor lattice trees and lattice animals in dimensions $d\geq \dmintree$ and $d\geq \dminanimal$, respectively, the square condition holds, i.e., $\square(z_c)<\infty$. As a result, the susceptibility and radius of gyration critical exponents $\gamma$ and $\nu$ exist in the bounded-ratio sense and takes on the mean-field values $\gamma=1/2$ and $\nu=1/4$.

We next move to some further extensions of our results, starting with sharp asymptotics of the Fourier transform of the critical two-point function $\hat{\bar{G}}_{z_c}(k)$ for $k$ small:

For nearest-neighbor LTs and LAs in dimensions $d\geq \dmintree$ and $d\geq \dminanimal$, respectively, there exists a constant $\bar{A}(d)$ such that, as $|k|\rightarrow 0$,

We next investigate the asymptotics in $x$-space of $\bar{G}_{z_c}(x)$ for $x$ large, using the results of Hara in [15]:

For nearest-neighbor LTs and LAs in dimensions  $d\geq 27$,
and with the constant $\bar{A}(d)$ as in Theorem <ref>, as $|x|\rightarrow \infty$,

The restriction $d\geq 27$ originates as a condition in [15], where it is used to start a recursion in <cit.>. It is unclear to us whether this restriction can be avoided. We can, however, prove the convergence of the classical lace expansions for $d\geq \dmintree$ for LTs and $d\geq \dminanimal$ for LAs. Whether this is enough to obtain the $x$-space asymptotics in Theorem <ref> is open.

In Section <ref>, we give an overview of the proof of Theorem <ref>. We next discuss relations to the literature.

§.§ Relations to the literature

Critical exponents. For spread-out lattice trees and animals, as well as for nearest-neighbor lattice trees in sufficiently high (unspecified) dimensions, the statements that $\gamma=1/2$ and $\nu=1/4$ have been improved to fixed $n$, as in <cit.>. This means that there exist constants $A_\gamma, A_\nu $such that

∑_x∈ t_n(x)=A_γ n^γ-1(1+o(1)),
R_n =A_ν n^ν(1+o(1)).

Here, the critical exponent $\gamma-1$ is sometimes called $\theta$. Such results are proved by using Tauberian Theorems on $\chi(z)=\sum_{n\geq 0} \sum_{x\in\Zd} t_n(x)z^n$ and $\sum_{n\geq 0}\sum_{x\in\Zd} |x|^2 t_n(x) z^n$, where $z\in \mathbb{C}$ with $|z|<z_c$. This requires careful Taylor expansions of these generating functions, with explicit control over the error terms. Possibly our results could be extended in this direction as well, but this would require a more detailed and extensive study of the analytic properties of $\chi(z)$ and $\hat{G}_z(k)$, which we refrain from.

Other interesting critical exponents exist. For example, the correlation length $\xi(z)$ is defined as

ξ(z)^-1=-lim_x→∞1/|x| logG_z(x).

It is predicted that $\xi(z)$ scales similarly as $\xi_2(z)$, so that, in particular,

ξ(z) ∼(z_c-z)^-1/4,

corresponding to $\nu=1/4$ also for this correlation length.

Asymptotics of $z_c$ and $g_{z_c}$.
The asymptotic value of $z_c$ and $g_{z_c}=\bar G_{z_c}(0)$, are shown in [34] to be given by

\begin{align}
\lbeq{conj-zc}
z_c&=\e^{-1}\left[\frac 1 {2d} +\frac {\frac 3 2}{(2d)^2}+\frac {\frac {115}{24}- \indAnimal\frac 1 2 \e^{-1}}{(2d)^3}\right]+o\big((2d)^{-3}\big),\\
g_{z_c}&=\e\left[\frac 1 {2d} +\frac {\frac 3 2}{(2d)^2}+\frac {\frac {263}{24}- \indAnimal\frac 1 2 \e^{-1}}{(2d)^2}\right]+o\big((2d)^{-2}\big),
\lbeq{conj-gzc}
\end{align}

where the factor $\indAnimal$ is $1$ for LA and $0$ for LT. The rigorous bound on the error terms was derived using the lace expansion. These asymptotic expansions have also been studied extensively in the physics literature, where results were obtained up to the order of $1/(2d)^6$, but with non-rigorous estimates on the error term. We refer the reader to [34] for an overview of such results and a discussion of the $1/d$ expansions of the critical value $z_c$ of SAW, percolation, LT, and LA.

Relations to super-processes. There has been a considerable effort to identify super-process limits of lattice trees and lattice animals, with a focus on lattice trees. Derbez and Slade [7, 8] show that the scaling limits of lattice trees of a fixed size scale to a finite measure called integrated super-Brownian excursion (ISE). For such a scaling limit, a lattice tree of size $n$ is seen as a random counting measure $X_n$, where $X_n(E)$ is the number of vertices of a random lattice tree that are in $E\subseteq \R^d$. When rescaling space by $n^{1/4}$ (which is related to $\nu=1/4$ in Corollary <ref>), and rescaling mass by $1/n$ (so that the total mass becomes 1, this random measure converges to ISE. Holmes and collaborators, instead consider the super-Brownian motion limits of lattice trees, proving convergence in finite-dimensional distributions  [27, 23] (see also [28]) and tightness [24], in the context of spread-out lattice trees above 8 dimensions. Extensions of such results to nearest-neighbor lattice trees, as well as to lattice animals, are of great interest.

§ OVERVIEW OF THE PROOF

In this section, we give a brief overview of how we derive our main results. This overview is similar in spirit to that in <cit.>, where nearest-neighbor percolation was investigated. We repeat part of this discussion, as it provides the key ideas in our proof.

§.§ Philosophy of the proof

We reduce the proof of Theorem <ref> to three key propositions and a computer-assisted proof. These ingredients involve

* the derivation of the non-backtracking lace expansion (NoBLE) in Proposition <ref>;

* the diagrammatic bounds on the NoBLE coefficients in Proposition <ref>;

* the analysis presented in [12] to obtain the infrared bound in Theorem <ref> for all $z\leq z_c$, as stated in Proposition <ref>; and

* a computer-assisted proof to verify the numerical conditions arising in the analysis in [12].

The philosophy behind the proof of these parts is discussed in Sections <ref>-<ref>, respectively. In Sections <ref> and <ref> we prove parts (a) and (b), respectively. In Section <ref>, we explain how we obtain part (c) using the analysis of [12].
The computer-assisted proof of part (d) is performed in [9] using the results of this paper.
For the analysis in the generalized setting [12] we state assumptions, which we verify in  Section <ref> and in Section <ref>.
Part (d) is explained in detail in Section <ref>, where we describe how the necessary computations are performed in several Mathematica notebooks. The mathematics behind the notebooks is explained in [12].

Let us now explain how our main results are proved. The notebooks contain a routine that verifies whether the numerical assumption on the expansions are satisfied and thereby whether the analysis of [12] yields the infrared bounds for a given dimension. The results in [12], combined with a successful verification using the notebooks, thus prove Proposition <ref>, which in turn implies Theorem <ref>. See also Figure <ref> for a visual description of the proof of Theorem <ref>.  Theorem <ref> will follow from the numerical bounds derived in the proof. The proofs of Corollary <ref>, Theorem <ref> and Theorem <ref> are completed in Section <ref>, using the results of Hara [15], as well as an improvement of the analysis there by Hara that we learned about in private communication [16], which yields an improved numerical condition for the convergence of the classical-lace expansion coefficients, allowing us to conclude the $x$-space asymptotics in Theorem <ref> for $d\geq 27$. We close this section with a discussion of our method and results.

[->,line width=2] (2.5,0) to (6.25,0);
[->,line width=2] (3.75,-4.75) to (6.25,-4.75);
[<-,line width=2] (10,-1.25) to (10,-3);
[->,line width=2] (9.5,-1.25) to (9.5,-3);
[->,line width=2] (0,-1.25) to (0,-3.25);

[<-,line width=2] (9.75,-5.75) to (9.75,-6.75);

[above]   at(4.4,0.1)     General relation;
[below]   at(4.6,-5)   Bounds in;
[below]   at(4.6,-5.5)    form of diagrams;

[left]   at(0,-1.5)        Coefficients to;
[left]   at(0,-2)        describe the;
[left]   at(0,-2.5)        perturbation;
[right]   at(10,-1.75)      Bound on the;
[right]   at(10,-2.25)      perturbation;

[left]   at(9.5,-1.5)      Prior bounds;
[left]   at(9.5,-2)       on the two-point ;
[left]   at(9.5,-2.5)      function;
[right]   at(10,-6)      Numerical bounds;
[right]   at(10,-6.5)      on diagrams;

Proposition <ref>

[draw]  at (10,0)

Analysis/ Bootstrap

in accompanying paper [12]

Model independent

[draw]  at (10,-4.5)

Numerical bounds

in accompanying

Mathematica notebooks [9]

[draw]  at (10,-8)

Numerical computation

in accompanying

Mathematica notebooks [9]

[draw]  at (0,-4.5)

Diagrammatic bounds

Propostion <ref>

Structure of the non-backtracking lace expansion.

In the following, we explain the philosophy behind the non-backtracking lace expansion (NoBLE), and start by describing simple random walk and non-backtracking walk:

Simple random walk.
An $n$-step nearest-neighbor simple random walk (SRW) on $\Zd$ is an ordered $(n+1)$-tuple $\omega=(\omega_0,\omega_1,\omega_2,\dots, \omega_n)$, with $\omega_i\in\Zd$ and $|\omega_i-\omega_{i+1}|_1=1$, where $|x|_1=\sum_{i=1}^d |x_i|$. Unless stated otherwise, we take $\omega_0=(0,0,\dots,0)$. Then, for $n\geq 1$,

p_n(x)   =#{$n$-step SRWs with $\omega_n=x$} =∑_y∈ 2d D(y)p_n-1(x-y)

=2d (D ⋆p_n-1)(x)=(2d)^n D^⋆n(x),SRWRecScheme

with $D$ being defined in def-Dhat and $f^{\star n}$ denotes the $n$-fold convolution of a function $f$.
The SRW two-point function is defined by

\begin{eqnarray}
\lbeq{genSRW}
C_z(x)&=&\sum_{n=0}^\infty p_n(x)z^n,\qquad\text{and}
\qquad \hat C_z(k) =\frac {1}{1-2dz\hat D(k)}
\end{eqnarray}

for $|z|<1/(2d)$, in $x$-space and $k$-space, respectively. The SRW susceptibility is given by

\begin{eqnarray}
\chi^{\sss\rm SRW}(z)= \hat C_z(0)= \frac {1} {1-2dz},
\end{eqnarray}

for $|z|<z_c$, with critical point $z_c=1/(2d)$.

Before introducing non-backtracking random walk, we introduce some notation that will be used throughout this document. We exclusively use the Greek letters $\iota$ and $\kappa$ for values in $\{-d,-d+1,\dots,-1,1,2,\dots,d\}$ and denote the unit vector in direction $\iota$ by $\ve[\iota]\in\Zd$, e.g. $(\ve[\iota])_i=\text{sign}(\iota) \delta_{|\iota|,i}$. We use ${\mathbb C}^{2d}$-valued and ${\mathbb C}^{2d}\times{\mathbb C}^{2d}$-valued quantities. For a clear distinction between scalar-, vector- and matrix-valued quantities, we always write ${\mathbb C}^{2d}$-valued functions with a vector arrow (e.g. $\vec v$) and matrix-valued functions with bold capital letters (e.g. ${\bf M}$). We do not use $\{1,2,\dots,2d\}$ as the index set for the elements of a vector or a matrix, but use $\{-d,-d+1,\dots,-1,1,2,\dots,d\}$ instead. We denote the identity matrix by $\mI\in{\mathbb C}^{2d\times 2d}$ and the all-one vector by $ \v1 =(1,1,\dots,1)^T\in{\mathbb C}^{2d}$. Moreover, we define the matrices $\mJ,\mD[k]\in{\mathbb C}^{2d\times 2d}$ by

\begin{eqnarray}
\lbeq{J-D-matrix-def}
(\mJ)_{\iota,\kappa}=\delta_{\iota,-\kappa}\qquad\text{ and }\qquad
(\mD[k])_{\iota,\kappa}=\delta_{\iota,\kappa} \e^{\ii k_\iota},
\end{eqnarray}

where $k\in[-\pi,\pi]^d$ and for negative index $\iota\in\{-d,-d+1,\dots,-1\}$, we write $k_\iota=-k_{|\iota|}$. Now we can introduce non-backtracking random walk and its Green's function.

Non-backtracking walk.
If an $n$-step SRW $\omega$ satisfies $\omega_i\not=\omega_{i+2}$ for all $i=0,1,2,\dots,n-2$, then we call $\omega$ a non-backtracking walk (NBW).  The NBW two-point function $B_z$ is defined, for $|z|<1/(2d-1)$, by

\begin{align*}
B_{z}(x)&=\sum_{n=0}^{\infty} \#\{\text{$n$-step NBW with $\omega_n=x$}\}z^n.
\end{align*}

As derived in <cit.>, this two-point function satisfies

\begin{eqnarray}
\lbeq{NBWGen}
\hat B_{z}(k)&=& \frac {1} {1 -z\v1^T\left[\mD[k]+z \mJ\right]^{-1}\v1}=\frac {1}{1-2d z\frac {\hat D(k)-z}{1-z^2}}.
\end{eqnarray}

The critical NBW and SRW two-point functions are thus related by

=2d-2/2d-1 ·1/1-D̂(k).

This link allows us to compute values for the NBW two-point function in $x$- and $k$-space, using the SRW two-point function. A detailed analysis of the NBW including a proof that the NBW, when properly rescaled, converges to Brownian motion can be found in [11].

§.§ Part (a): Non-Backtracking Lace Expansion (NoBLE)

In this section, we explain the shape of the Non-Backtracking Lace Expansion (NoBLE), which is a perturbative expansion of the two-point function. For this, we derive an equation alike NBWGen for the two-point function $G_z(x)$. This expansion is derived in Section <ref>.

Next to the usual two-point function $G_z$, see defLTLATwoPoint,  we use a slight adaptation $G_z^\iota$, with $\iota\in\{\pm1,\pm2,\dots, \pm d\}$ being a distinct direction. We postpone the definition of $G_z^\iota$ to Section <ref>, see defGzIotaanimal. Intuitively, $G_z^\iota$ considers only LTs and LAs that do not contain $e_\iota$, but for technical reasons, the precise definition is somewhat more involved. Our analysis relies on two expansion identities relating $G_z(x)$ and $G_z^\iota(x)$ that are formalized in the following proposition:

For every $x\in \Z^d$, $\iota,\kappa\in \{\pm1,\pm2, \dots, \pm d\}$, the following recursion relations hold:

G_z(x)   =δ_0,x+∑_y∈, κ∈{±1,…,±d} (δ_0,y+Ψ_z^κ(y))G^κ(x-y+[κ])+Ξ_z(x),

G_z(x)   =G^ι_z(x)+G^-ι_z( x-[ι])
+∑_y∈, κ∈{±1,…,±d} Π_z^ι,κ(y) G_z^κ(x-y+[κ])+Ξ_z^ι(x),

Π_z^ι,κ(x)    = ∑_N=0^∞ (-1)^N    Π^[N],ι,κ(x),
Ξ_z(x)= ∑_N=0^∞ (-1)^N Ξ^[N]_z(x),

Ψ_z^κ(x)   =∑_N=0^∞ (-1)^N Ψ^[N],κ_z(y),
Ξ_z^ι(x)=∑_N=0^∞ (-1)^N Ξ^[N],ι(x),

=  zg^ι_z,                    =  zg_z.

Explicit formulas for the above lace-expansion coefficients are given in Section <ref>.

Equations Gx-in-ex-1 and Gx-in-ex-2 are similar in spirit to the equations satisfied by the NBW two-point function, where $G_z(x)$ is replaced by $B_z(x)$ and $G^\iota_z(x)$ by $B^\iota_z(x)$, which is the Green's function of all NBWs for which the first step is unequal to $e_{\iota}$ (recall [12]). For this, the similar equation to Gx-in-ex-1 is obtained by conditioning on the first step, while Gx-in-ex-2 is obtained by splitting depending on whether the first step equals $e_{\iota}$ or not. Equations Gx-in-ex-1 and Gx-in-ex-2 reduce to the equations for NBW when we set $\Psi^{\kappa}_z=\Xi_{z}=\Pi_{z}^{\iota,\kappa}\equiv 0$, and replace $\aaz$ by $(2d-1)z$ and $\aabz$ by $2dz$. Thus, we can think of  Gx-in-ex-1 and Gx-in-ex-2 as a perturbations around NBW, and of $\Psi^{\kappa}_z=\Xi_{z}=\Pi_{z}^{\iota,\kappa}$ as the perturbation coefficients, of which we hope to prove that they are `small' in an appropriate sense.

Of course, the precise formulas for the lace-expansion coefficients are crucial for our analysis to succeed. However, at this stage, we refrain from stating their forms explicitly, and refer to Section <ref> instead where they are derived while performing the NoBLE.

Applying the Fourier transforms to Gx-in-ex-1 and Gx-in-ex-2 and some simple rearrangements, see <cit.>, we derive that

\begin{align}
\lbeq{lace-exp-eq}
\hat{G}_z(k)=\frac{1+\hat{\Xi}_{z}(k)-\aaz(\v1+\vPsi[k])^T\big[\mD[k]+\aaz\mJ  +\mPi[k]\big]^{-1} \vXi[k]}
{1-\aaz(\v1 +\vPsi[k])^T\big[\mD[k]+\aaz \mJ+\mPi[k]\big]^{-1}{\v1}},
\end{align}

(Ψ⃗̂⃗(k))_κ=Ψ̂^κ_z(k),    ([k])_ι=Ξ̂^ι_z(k),     ([k])_ι,κ=Π̂^ι,κ_z(k),

Equation lace-exp-eq is the NoBLE equation, and is the workhorse behind our proof. The goal of the NoBLE is now to show that lace-exp-eq is indeed a perturbation of NBWGen. This amounts to proving that $\hat{\Xi}_{z}(k)$,$\vXi[k]$, $\vPsi[k]$ and $\mPi[k]$ are small, which will only be true in sufficiently high dimensions.

We continue by discussing how to bound the NoBLE coefficients.

§.§ Part (b): Bounds on the NoBLE

The content of the second key proposition is that the NoBLE coefficients can be bounded by combinations of simple diagrams. Simple diagrams are combinations of two-point functions, alike the following examples:

\begin{align}
(2d\aaz)^2 (G_z\star D\star D\star G_z)(e_\iota),   \quad
\sup_{x\in\Zd\colon \ |x|^2>2}  \sum_{y\in\Zd} |y|^2G_z(y) (G_z\star D)(x-y).
\end{align}

For each $N\geq 0$, the NoBLE coefficients
$ \Pi^{\ssc[N],\iota,\kappa}(y),$ $\Xi^{\ssc[N]}(x),$ $\Psi^{\ssc[N],\kappa}(y)$ and $\Xi^{\ssc[N],\iota}(x)$  can be bounded by a finite combination of sums and products of simple diagrams.

The explicit form of the bounds in Proposition <ref> is given in Lemmas <ref>-<ref>. In Section <ref> we summarize the bounds as required for the analysis in [12]. As the complete proof of the bounds on the NoBLE coefficients is highly elaborate, we only sketch the proof.
We describe the used diagrams used in the bounds in Section <ref> and give the formal definition in the appendix.
In this paper, we only informally define the diagrams that we use to bound the coefficients.
A complete definition and additional details can be found in the extended version [14].

Matrix-valued bounds.
The lace-expansion coefficients arising in the NoBLE describe contributions created by multiple mutually intersecting paths, in the LTs and LAs. We will informally call such path-intersections loops, and this notion will be made precise below. In the NoBLE, these loops require at least $4$ bonds by design, as direct reversals are excluded by the way that we set up the NoBLE expansion. These lace-expansion coefficients can be bounded in terms of certain Feynman diagrams, of which the lines correspond to various two-point functions, and these lines are required to have various intersections. We will refer to the two-point functions arising in these bounds informally as `lines'.

The diagrams turn out to have the general property that lines can be part of at most two loops. To optimally use the information that loops contain at least 4 bonds, we distinguish five cases for the length and function of paths used by two loops.
Then, we bound the contribution of each loop of the lace-expansion diagram one-by-one, using the information on the lines shared with the previous and preceding loops. We explain this in detail in Section <ref>. This procedure naturally gives rise to a bound on the NoBLE coefficients in terms of matrix products, as formalized in Lemma <ref>. For example, our proof yields that

\begin{align*}
\sum_{x\in\Zd}\Xi^{\ssc[N]}_z(x)&\leq \frac {g_z}{\gj} \vec P^T{\bf A}^{N-2}{\vec E}^{\rm{open}},
\end{align*}

for $N\geq 3$, see Bound-Xi2, for certain vectors $\vec P, {\vec E}^{\rm{open}}$ and 5 by 5 matrices ${\bf A}$. We will give an interpretation of the elements in these vectors and matrices in Section <ref>. For our analysis, we require a bound on this when summed over $N$. To compute this bound numerically, we perform an eigenvector decomposition of $\vec P$, in terms of the eigenvectors $(\vec v_i)_{i=1}^5$ of ${\bf A}$ with corresponding eigenvalues $(\lambda_i)_{i=1}^5$. In this decomposition, we write $\vec P=\sum_{i=1}^5 \vec v_i,$ so that the eigenvectors used are not normalized[We do account for the possibility that ${\bf A}$ is not diagonalizable.].

\begin{align*}
\hat \Xi^{\ssc[N]}_{z}(0)\leq& \sum_{i=1}^5 \vec v_i \lambda^{N-2}_i{\vec E}^{\rm{open}}.
\end{align*}

The sum of this over $N$ is computed using the geometric sum, see <cit.> for more details.

The order of this bound is to a large extent given by the largest eigenvalue of ${\bf A}$. For example, for LTs in $d=16$,

\begin{align*}
{\bf A}=\left(
\begin{array}{ccccc}
0.0325769 & 0.0151778 & 0.0209234 & 0.0151778 & 0.0325769 \\
0.0196345 & 0.00927948 & 0.0131029 & 0.00927948 & 0.0196345 \\
0.0271802 & 0.0143934 & 0.0201978 & 0.0160374 & 0.0313802 \\
0.0196345 & 0.00927948 & 0.0131029 & 0.00927948 & 0.0196345 \\
0.0325769 & 0.0151778 & 0.0209234 & 0.0151778 & 0.0325769 \\
\end{array}
\right)
\end{align*}

with largest eigenvalue $\lambda_1=0.103089$. In the classical lace expansion, the corresponding $N$th lace-expansion diagram bound also decays exponentially in $N$, where the base of the exponent is a non-trivial square. We can bound the non-trivial square, as it arises in the classical lace expansion, by $2.53461$.

Revisiting the classical lace expansion and then using all our improvements, which we explain in Section <ref>, we can improve the base of the exponent to a value of $0.130586$.

Using our matrix-valued bounds, reduce the exponent by $30\%$. The gain in the explicite contribution is usually even larger.

§.§ Part (c): The NoBLE analysis

At the center of our analysis are the so-called bootstrap functions, which we next discuss.

Bootstrap functions.

For the bootstrap, we define the following functions:

\begin{eqnarray}
\lbeq{defFunc1}
f_1(z)&:=&\max\left\{(2d-1)zg_z,c_\mu (2d-1)zg_z^\iota\right\},\\
\lbeq{defFunc2}
f_2(z)&:=& \sup_{k\in(-\pi,\pi)^d}
\frac{\hat G_z(k)}{\hat B_{1/(2d-1)}(k)} = \frac{2d-1}{2d-2}\sup_{k\in(-\pi,\pi)^d} [1-\hat D(k)]\ \hat G_z(k),\\
\lbeq{defFunc3}
f_3(z)&:=& \max_{\{n,l,S\}\in \mathcal{S}} \frac{\sup_{x\in S}
\sum_{y}|y|^2G_z(y)(G_z^{\star n}\star D^{\star l})(x-y)}{c_{n,l,S}},
\end{eqnarray}

where $c_\mu>1$ and $c_{n,l,S}>0$ are some well-chosen constants and $\mathcal{S}$ is some finite set of indices. Let us now discuss the choice of these functions, and explain their use.

The functions $f_1$ and $f_3$ can been seen as combinations of multiple functions.  We group these functions together as they play a similar role and are analyzed in the same way. We do not expect that the values of the bounds on the individual functions that constitute $f_1$ and $f_3$ are comparable. This is the reason that we introduce the constants $c_\mu$ and $c_{n,l,S}$.

The choice of point-sets $S\in \mathcal{S}$ improves the numerical accuracy of our method considerably. For example, we obtain much better estimates in the case when $x=0$ (leading to a closed diagram), than for $x\neq 0$. For $x$ being a neighbor of the origin, we can further use symmetry to improve our bounds significantly. To obtain the infrared bound as stated in Theorem <ref>, we use the following set $\mathcal{S}$:

\begin{align}
\lbeq{calS-def}
\mathcal{S}=\big\{
&\qquad \{2,0,\mathcal{X}\},\{2,1,\mathcal{X}\},\{2,2,\mathcal{X}\},\{2,3,\mathcal{X}\},\{2,6,\{0\}\}\big\},\nn
\end{align}

with $\mathcal{X}=\{x\in\Zd\colon |x|_2>1\}$. This turns out to be sufficient for our main results.

We apply a forbidden region or bootstrap argument that is based on three claims:

(i) $z\mapsto f_i(z)$ is continuous for all $z\in [z_{I}, z_c)$ and $i=1,2,3$ and for some $z_{I}\in(0,z_c)$;

(ii) $f_i(z_{I})\leq \gamma_i$ holds for $i=1,2,3$; and

(iii) if $f_i(z)\leq \Gamma_i$ holds for $i=1,2,3$, then, in fact, also $f_i(z)\leq \gamma_i$ holds for every $i=1,2,3$, where $\gamma_i<\Gamma_i$ for every $i=1,2,3$.

Together, these three claims imply that $f_i(z_{I})\leq \gamma_i$ holds for every $i=1,2,3$ and all $z\in [z_{I}, z_c)$. This in turn implies the statement of Theorem <ref>  for all $z\in [z_{I}, z_c)$.

The continuity in Claim (i) is proven in [12] under some assumption that we explain and prove  below. The proofs of the initialization of the bootstrap in Claim (ii) as well as the improvement of the bounds in Claim (iii) use the following relations that are also sketched in Figures <ref> and <ref>:

* simple diagrams can be bounded by a combination of two-point functions, see <cit.>;

* the NoBLE coefficients can be bounded by a combination of simple diagrams, that, in turn, can be bounded using the bounds on the two-point functions that we obtain from the bootstrap assumption, see Section <ref>;

* bounds on the NoBLE coefficients imply bounds on the two-point function, see <cit.>, which allows us to improve upon the bounds that we assumed.

Thus, whenever we have numerical bounds on simple diagrams, or on NoBLE coefficients, or on the two-point function, we can also conclude bounds on the other two quantities.

We choose $z_{I}$ such that $G_{z_I}(x)\leq B_{1/(2d-1)}(x)$ holds pointwise in $x$.
Then, using that $G_{z_I}(x)\leq B_{1/(2d-1)}(x)$ and that we can compute $B_{1/(2d-1)}(x)$ numerically, we verify the initialization of the bootstrap in Claim (ii) (i.e., $f_i(z_I)\leq \gamma_i$ for $i=1,2,3$) numerically, see Figure <ref>.

[line width=1pt,auto,scale=0.7]
at (3*cos(153),3*sin(153))     $f_i(z_I)\leq \gamma_i$;
at (3*cos(0),3*sin(0))      Bounds on simple diagrams;
at (3*cos(230),3*sin(230)+0.2)      Bounds on coefficients;
[->,very thick] (3*cos(215)-0.2,3*sin(215))  arc  (215:165:3);

[->,very thick] (3*cos(350),3*sin(350))  arc  (350:240:3);
[->,very thick] (4,2.2) to (3,0.5);

at(4,2.5)      $G_{z_I}(x)\leq B_{1/(2d-1)}(x)$;
[left]   at(-3.1,0)   Conclude a bound;

Initialization of the bootstrap: proof that $f_i(z_I)\leq \gamma_i$ holds for $i=1,2,3$. Here $\gamma_1,\gamma_2,\gamma_3$ are appropriately and carefully chosen constants.

The proof of Claim (iii) is the most elaborate step of our analysis. Its structure is shown in Figure <ref>. We start from the assumption that $f_i(z)\leq \Gamma_i$ holds for every $i=1,2,3$. The function $f_1$ gives a bound on $zg_z$ and $zg_z^\iota$. The function $f_2$ allows us to bound the two-point function in Fourier space by $\hat B_{1/(2d-1)}(k)$, which we can integrate numerically to obtain numerical bounds on simple diagrams. Further, $f_2$ allows us to conclude the infra-red bound in Theorem <ref>. These, in turn, imply bounds on the NoBLE coefficients, which we use to improve our bounds on the bootstrap functions.

In the case that the computed bounds are small enough, we can conclude that $f_i(z)\leq \gamma_i$ holds and thereby that the improvement of the bounds in Claim (iii) holds. Whether we can indeed prove that Claim (iii)  holds depends on the dimension we are in, the quality of the bounds and the analysis used to conclude bounds for the bootstrap function. In high enough dimensions (e.g. $d\geq 1000$) the perturbation is rather small so that it is
relatively easy to prove Claim (iii). Proving the claim in lower dimensions is only possible  when the bounds on the lace-expansion coefficients and the analysis are sufficiently sharp and hence sophisticated. It is here that it pays off to apply the NoBLE compared to the classical lace expansion. The third step in the proof of Theorem <ref> is formalized in the following proposition:

[line width=1pt,auto,scale=0.7]
at (3*cos(90),3*sin(90))     $f_i(z)\leq \Gamma_i$;
at (3*cos(153),3*sin(153))     $f_i(z)\leq \gamma_i$;
at (3*cos(0),3*sin(0))      Bounds on simple diagrams;
at (3*cos(230),3*sin(230)+0.2)      Bounds on coefficients;
[->,very thick] (3*cos(215)-0.2,3*sin(215))  arc  (215:165:3);
[->,very thick] (3*cos(70),3*sin(70))  arc  (70:10:3);
[dotted,very thick] (3*cos(140),3*sin(140))  arc  (140:115:3);
[->,very thick] (3*cos(350),3*sin(350))  arc  (350:240:3);
[->,very thick] (-4,3) to (3*cos(90)-1.4,3*sin(90)) ;

[left]   at(-4,3)      Assume a bound;
[left]   at(-3.1,0)   Conclude a bound;

Proof of claim (iii): $f_i(z)\leq \Gamma_i$ for $i=1,2,3$ implies that $f_i(z)\leq \gamma_i$ for $i=1,2,3$.

For nearest-neighbor LTs and LAs, in  $d\geq \dmintree$ and $d\geq \dminanimal$, the NoBLE analysis of [12] applies and proves the infrared bound in Theorem <ref>. In particular, there exist constants $\Gamma_1,\Gamma_2,\Gamma_3$ and $\gamma_1,\gamma_2,\gamma_3$ such that $f_i(z_I)\leq \gamma_i$ for $i=1,2,3$ and, for every $z\in [z_I,z_c)$, the bounds $f_i(z)\leq \Gamma_i$ for $i=1,2,3$ imply that $f_i(z)\leq \gamma_i$ for $i=1,2,3$.[In the percolation paper, we erroneously wrote $p<p_c$ instead, but also there we need that $p\in[p_I,p_c)$.]

As shown in Figure <ref>, Proposition <ref> is proved using the results of Propositions <ref>-<ref>, the analysis of [12] and the computer-assisted proof performed in the Mathematica notebook that can be found on [9].  To apply the general NoBLE analysis of [12], we need to prove that the assumptions formulated in [12] hold.
All but one of these assumptions are proven in Section <ref>.
The one remaining assumption, involving the bounds on the NoBLE coefficients, is addressed in Section <ref>.

§.§ Part (d): Numerical analysis

In this section, we explain how the numerical computations are performed using Mathematica notebooks that are available from the first author's homepage [9].

Simple-random walk computations.
The procedure starts by evaluating the notebook .
The file computes the value of the SRW, and thereby the NBW, two-point function for various locations in $\Zd$.
This computation uses pre-computed values of the number SRWs paths and SRW integrals based on numerical integration of certain Bessel functions.
These computations rely on ideas from <cit.>, and are explained in detail in <cit.>.
The SRW integrals provide rigorous numerical upper bounds on various convolutions of SRW Green's functions with themselves, evaluated at various $x\in \Z^d$. We rely on 140 of such integrals.

Running these programs takes several hours. For this reason, once computed, the results are saved in two files,  and  and are loaded automatically when the notebook is evaluated a second time for the same dimension.
Alternatively, these two files can also be downloaded directly from the home page of the first author, and put in one's own home directory.[In Mathematica, the command $InitialDirectory will tell you what this directory is.]

Implementation of the NoBLE analysis for LT and LA.
After having computed all the simple random walk ingredients, we evaluate the notebook , that implements the bounds of the NoBLE analysis [12]. After this, we are ready to perform the NoBLE analysis for the model by evaluating the notebook  and , respectively. In these notebooks, we implement all the bounds proven in this paper. The computations in ,  and  merely implements the bounds proven in this paper and in [12], and rely on many multiplications and additions, as well as the diagonalization of two 5-by-5 matrices. These computations could in principle be done by hand (even though we prefer a computer to do them).

Output of Mathematica Notebooks. After having evaluated the Mathematica notebooks, we can verify whether the analysis has worked with the chosen constants $\Gamma_1,\Gamma_2, \Gamma_3$.
In Figure <ref> shows the relevant part of the output of the  notebook.
Let us explain this output in more detail. The green dots mean that the bootstrap has been successful for the parameters as chosen. When evaluating the notebook, it is possible that some red dots appear, and this means that these improvements were not successful. The first 3 dots in the first table are the verifications that $f_i(z_I)\leq \gamma_i$ for $i=1,2,3$. The next three dots show that the improvement has been successful for any $z\in(z_I,z_c(d))$. The values for $\Gamma_1, \Gamma_2,\Gamma_3$ are indicated in the second row.
For example, $\Gamma_1=1.10225007$ means that we assumed that $f_1(z)=(2d-1)zg_z\leq1.10225007$.
Using this assumption we concluded that $f_1(z)\leq 1.102250067$ from the improvement step (which relies on the NoBLE analysis), so that we can choose  $\gamma_1=1.02250069$ for our analysis in Proposition <ref>. Since this is true for all $z\in(z_I,z_c(d))$, we obtain that $(2d-1)z_cg_{z_c}\leq 1.0225007$. This explains the value in the table of Theorem <ref>.
The stated upper bound on $g_{z_c}$ follow from $g_{z_c}\leq \e\Gamma_1$, which we obtain by combining $f_1(z)<\Gamma_1$ with $z_c>(2d-1)^{-1}\e^{-1}$, which is proven in Section <ref>.

Similarly $\Gamma_2=1.335307$ implies that

\begin{align}
&\frac{2d-1}{2d-2}\sup_{k\in(-\pi,\pi)^d} [1-\hat D(k)]\frac 1 {g_z}\hat {\bar G}_z(k)\leq \Gamma_2,
\end{align}

which, in turn, implies that

\begin{align}[1-\hat D(k)]\hat {\bar G}_z(k) \leq g_z\frac {2d-2}{2d-1}\Gamma_2
\leq g_z\frac {2d-2}{2d-1}\Gamma_2\leq \Gamma_1\Gamma_2\e\leq 3.955 :=\bar{A}_2(d).
\end{align}

Using these computation, we have computed the bounds stated in Theorem <ref>.
Anyone interested in obtaining improved bounds on $g_{z_c}$, $g_{z_c}z_c$ or $A_2$ for other values of $d$ can play with the notebooks to optimize them. The second and third table in Figure <ref> provides  details on the improvement of $f_3(z)$, which, as indicated in defFunc3 and calS-def consists of several contributions, over which the maximum is taken. The assumed bound correspond to the constants $c_{n,l,S}$, with $S\in {\mathcal S}$ in calS-def. The notebooks  and  also includes a routine that optimizes the choices of $\Gamma_i$ and $c_{n,l,S}$. This makes it possible to efficiently find values for which the analysis works (when these exist).

Output of the Mathematica notebook , where we can see that the bootstrap argument can be successfully initialized, as well as successfully improved. For completeness, we also show the various bounds of the components of $f_3$.

Output of the Mathematica notebook , where we can see that the bootstrap argument can be successfully initialized, as well as successfully improved.

§.§ Structure of the NoBLE proof and related results

Summary of the proof of the infrared bound in Theorem <ref>.
We have already seen how delicately the four parts of the proof described in Section <ref>
are intertwined. The expansion in part (a) provides a characterisation of $\hat G_z(k)$ as a perturbation of $\hat B_{1/(2d-1)}(k)$ involving the NoBLE coefficients. The analysis in part (c) allows us to compute bounds on $\hat G_z(k)$, when numerical bounds on the coefficients are available.  To obtain such bounds we need to derive diagrammatic bounds, as formulated in part (b), that bound the NoBLE coefficients by simple diagrams.  However, we rely on bounds on $G_z$ to bound such simple diagrams numerically. Thus, we obtain a circular reasoning.

Using the bootstrap argument we can complete the circle, see Figures <ref>-<ref>, to obtain a bound on $\hat G_z(k)$ for all $z\in [z_I, z_c)$. For the bootstrap argument, we need to show that $f_i(z_I)\leq \gamma_i$, as well as the fact that $f_i(z)\leq \Gamma_i$ implies that $f_i(z)\leq \gamma_i$, for appropriately chosen $\gamma_i$ and $\Gamma_i$ for all $z\in(z_I, z_c)$. The verification whether $f_i(z_I)\leq \gamma_i$ holds for $i=1,2,3$. Whether we can conclude from $f_i(z)\leq \Gamma_i$ for $i=1,2,3$ that also $f_i(z)\leq \gamma_i$ holds for $i=1,2,3$ requires a computer-assisted proof as indicated in Section <ref>. Starting from $G_{z_I}(x)\leq  B_{1/(2d-1)}(x)$, $f_i(z)\leq \Gamma_i$ for $i=1,2,3$ and explicit computations of $B_{1/(2d-1)}(x)$, we obtain numerical bounds on simple diagrams.  These are then used to obtain numerical bounds on the NoBLE coefficients, which we in turn use to verify whether we can actually conclude from $f_i(z)\leq \Gamma_i$ for $i=1,2,3$ that $f_i(z)\leq \gamma_i$ for $i=1,2,3$ holds.

Combining these steps yields the required results for $z\in [z_I, z_c)$. We obtain the statement also for $z=z_c$ by using that $\hat G_z(k)/\hat B_{1/(2d-1)}(k)$ and the NoBLE-coefficients are continuous and uniformly bounded for $z\in [z_I, z_c)$ as well as left-continuous in $x$-space at $z_c$.

Extension to all $d\geq 20$. Theorems <ref> and <ref> prove the infrared bound, as well as estimates on amplitudes and critical values, for $d=\dmintree, \ldots, 20$ for lattice trees, and for $d=\dminanimal, \ldots, 20$ for lattice animals. The Mathematica notebooks, and the numerical estimates that these rely upon, are designed to prove our results for any specific dimension. We successfully execute them for $d$ up to $29$.

To prove that the statements hold for all higher dimensions $d\geq 30$, without spending eternity checking the condition one dimension at time, we further modify our numerical verifications so that they only use quantities that are monotone decreasing in the dimension. It results in bounds on the NoBLE coefficients and related quantities that hold uniformly in all $d\geq 30$, implying that the bootstrap succeeds in all these dimensions in one go. This numerical verification is performed in the Mathematica notebook  for LA. This analysis also immediately applies to LTs, as all LT quantities are bounded by their LA analogues.

The idea behind these uniform bounds, used in the improvement of the bootstrap bounds, is the following:

* The SRW quantities that we rely upon, such as the Green's function at various points in $\Z^d$ and contributions of short paths, are monotone decreasing in $d$.

* Simple diagram are bounded using the bootstrap assumption $f_i(z)\leq \Gamma_i$ and these SRW quantities.
Using the same $\Gamma_i$ for all $d\geq 30$, the bounds on simple diagrams are monotone decreasing in $d$ as well.

* Our NoBLE coefficients are bounded using simple diagrams, which implies that these bounds are also monotone decreasing in $d$.

* Our bounds on the NoBLE coefficients decrease in $d$, so that the bounds on the coefficients appearing in the analysis of [12] are monotone decreasing as well. In the end, this guarantees that the improvement of the bootstrap bounds succeeds for all $d\geq 30$ at once.

In Appendix  <ref>, we explain how we modify our numerical verifications, show that the SRW quantities that we rely upon are monotone decreasing in $d$, and discuss the additional steps needed to make the analysis work for all $d\geq 30$.

Organization of this paper.
The remainder of this paper is organised as follows. In Section <ref>, we perform the NoBLE, and thus prove Proposition <ref>. There, we also give explicit formulas for the NoBLE coefficients. In Section <ref>, we explain how diagrammatic bounds on the NoBLE coefficients can be obtained. These diagrammatic bounds are phrased in terms of various building blocks that are informally defined in Section <ref>. In Section <ref>, we explain how such diagrammatic bounds are proved.

[line width=1pt,auto,scale=#1]

[right]   at(10,1)      $y$;
[below]   at(0,0)      $x$;
[below]   at(1,0)      $w$;
[above]   at(-1,1)      $S_0$;
at(5,-2)      Further, $S_3=S_4=S_6=S_7=S_9=\varnothing$.;
(10,1) circle (3pt);
(0,0) circle (3pt);

[-] (0,0) to  (1,0);
[below]   at(0.5,0)      $b_1$;
[-] (2,1) to  (4,1);
[below]   at(2.5,0)      $b_2$;
[below]   at(3.5,0)      $b_3$;
[-] (4,1) to  (4,2);
[right]   at(4,1.5)      $b_4$;
[-] (4,2) to  (5,2);
[below]   at(4.7,2)      $b_5$;
[-] (5,2) to  (5,0);
[right]   at(5,1.5)      $b_6$;
[right]   at(5,0.5)      $b_7$;
[-] (5,0) to  (6,0);
[below]   at(5.5,0)      $b_8$;
[-] (7,0) to  (7,1);
[right]   at(7,0.5)      $b_9$;
[-] (7,1) to  (8,1);
[below]   at(7.7,1)      $b_{10}$;

[draw] (3,2) to (0,2) to  (0,0) to  (-1,0) to  (-1,1) to  (-2,1);
[-] (1,2) to  (1,3);

[draw] (1,0) to  (1,1) to  (2,1) to  (2,0)to  (1,0);
[-] (2,0) to  (2,-1);

[draw] (1,0) to  (1,1) to  (2,1) to  (2,0)to  (1,0);
[-] (2,0) to  (2,-1);
- [above]   at(1.5,1)      $S_1$;

[above]   at(3,1)      $S_2=\varnothing$;
[-] (5,2) to  (6,2);
[above]   at(5.5,2)      $S_5$;

[above]   at(6.5,0)      $S_8$;

[draw]    (6,0)to  (6,-1)to  (7,-1)to  (7,0) to (6,0);
[-] (7,-1) to  (8,-1);

[draw] (8,1)to  (9,1)to  (9,2)to  (8,2) to  (8,1);
[draw] (9,0) to  (9,1) to  (10,1) to  (10,0) to (9,0) ;
[above]   at(9,2)      $S_{10}$;

A LA containing $x$ and $y$. All bonds of the backbone $(b_i)_{i=1,\dots,10}$ and all sausages $(S_i)_{i=0,\dots,10}$ are labeled in the picture.

§ THE NON-BACKTRACKING LACE EXPANSION

§.§ Derivation of the expansion

In this section we derive the NoBLE for LTs and LAs using an algebraic expansion. The basic idea is similar to the classical lace expansion applied to SAWs derived first in [6], and is an adaptation of the classical lace expansion for LTs and LAs derived in [20]. Let us consider a LT that contains $0,x$. As a LT does not contain any loops there exists a unique path connecting $0$ and $x$. We call this path the backbone. A LA containing $0,x$ can contain loops. Therefore, a connection between $0$ and $x$ is not necessarily unique. The analogue to the backbone for LAs is formed by the pivotal bonds, which are defined as follows:

Let $A$ be a LA that contains $x,y\in\Zd$. We say that $x$ and $y$ are doubly connected in $A$ if there exist two edge-disjoint paths $p_1,p_2\subset A$ connecting $x$ and $y$. By convention, we say that $x\in A$ is doubly connected to itself. We call a bond $b\in A$ pivotal for $x$ and $y$ if the removal of $b$ would disconnect $A$ into two disjoint animals, one containing $x$ and the other $y$.

[1]

Let us consider a LA containing $x$ and $y$ as given in Figure <ref>. Then, $x$ and $y$ are either doubly connected or there exists at least one pivotal bond $b$. If there are multiple pivotal bonds, then they have a natural order for the connection from $x$ to $y$, because every self-avoiding path from $x$ to $y$ has to use the pivotal bonds $(b_i)_i$ in the same order.

The removal of the bonds $(b_i)_{i\geq 1}$ disconnects the animal into mutually non-intersecting pieces, which we denote in Figure <ref> by $(S_i)_{i\geq 1}$. We will consider bonds to be directed, and write $b=(x,y)$ for the bond directed from $x$ to $y$. For a directed bond $b=(x,y)$, we write $\bar{b}=y$ and $\underline{b}=x$ for the two ends of the bond. The pieces $(S_i)_{i\geq 1}$ form double connections between the end of one pivotal bond $\bar b_i$ to the beginning of the following pivotal bond $\underline b_{i+1}$.
We call these doubly connected pieces $(S_i)_{i\geq 1}$ sausages, and the sequence of pivotal bonds $(b_i)_{i\geq 1}$ the backbone of the animal.
For LTs double connections are not possible, so that  $S_i=\{\bar b_i\}=\{\underline b_{i+1}\}$.

So far, we follow the classical lace expansion as in [20]. Now we start to deviate, and we wish to do so in a way that is closer to the NBW.
In order to do this, we define the rib-walk and sausage-walk to characterize the combination of a backbone and a set of ribs/sausages, that are non-backtracking.
Using these walks we derive the NoBLE for LTs and LAs, the steps of the classical lace expansion, by expanding a graph-based description of the avoidance constraint of the sausages and of the backbone.

* We call a LT $S$ that contains $x\in\Zd$ a rib for $x$ and define that the empty set is also a rib for all $x\in\Zd$ (recall that a LT is a collection of bonds).

* For $x,y\in \Zd$ and $n\geq 1$, we call a collection of $n$ oriented nearest-neighbor bonds $(b_i)_{i=1}^n$ and $n+1$ ribs $(S_i)_{i=0}^n$ an $n$-step rib-walk from $x$ to $y$, if $S_0$ is a rib for $\underline b_1=x$, $S_n$ is a rib for $\bar b_n=y$, and, for $i=1,\dots,n-1$, the LT $S_i$ is a rib for $\bar b_i=\underline b_{i-1}$. We call $(b_i)_{i=1}^n$ the backbone of the rib-walk.

* For a rib-walk $\omega=((b_i)_{i=1}^n, (S_i)_{i=0}^n)$, we define $|\omega|$ to be the number of bonds in the backbone. We denote the $i$th bond of the backbone by $b^{\omega}_i$  and the $i$th rib of $\omega$ by $S^{\omega}_i$.

* We say that any LT containing the origin is a zero step rib-walk to the origin.

* We call a rib-walk $\omega$ non-backtracking if $\bar b^\omega_{i+1}\nin S^\omega_{i},\underline b^\omega_{i}\nin S^\omega_{i}$ and $\bar b^\omega_{i+1}\neq \underline  b^\omega_{i}$ for all $i$.

* We define $\Wcal^\ssss[T](x)$ as the set of all rib-walks from the origin $0$ to $x$ and
$\Wcal^{\ssss[T],\iota}(x)$ to be the set of all rib-walks $\omega$ from $0$ to $x$ such that $\ve[\iota]\nin S^\omega_0$ and $\bar b^\omega_1\neq \ve[\iota]$.

We point out that a bond could be part of multiple ribs, so that there is no bijection between rib-walks and LTs containing $0$ and $x$.
This bijection is however possible if we restrict to self-avoiding rib-walks. The non-backtracking condition is part of this necessary self-avoidance constraint. It rules a specific notion of immediate reversals out. Thus, we can think of a LT as a non-backtracking rib-walk with extra mutual avoidance constraints between the ribs. We continue by defining similar quantities for LAs, to set the stage for an expansion that can treat LTs and LAs at the same time:

* We call a LA $S$ a sausage for $(x,y)\in\Zd\times\Zd$, if $x$ and $y$ are doubly connected in $S$.
Further, we define that the empty set is a sausage for every $(x,x)$ with $x\in\Zd$.

* For $x,y\in \Zd$ and $n\geq 1$ we call a collection of $n$ oriented nearest-neighbor bonds $(b_i)_{i=1}^n$ and $n+1$ sausages $(S_i)_{i=0}^n$ an $n$-step sausage-walk from $x$ to $y$, when $S_0$ is a sausage for $(x,\underline b_1)$, $S_n$ is a sausage for $(\bar b_n,y)$ and $S_i$ is a sausage for $(\bar b_i,\underline b_{i+1})$ for $i=1,\dots,n-1$.

* For a sausage-walk $\omega=((b_i)_{i=1}^n, (S_i)_{i=0}^n)$ we define $|\omega|$ to be the number of steps of $\omega$. We denote the $i$th bond of the backbone by $b^{\omega}_i$ and  the $i$th sausage of $\omega$ by $S^{\omega}_i$. We call $(b_i^\omega)_{i=1}^n=(b_i)_{i=1}^n$ the backbone of the sausage-walk.

* For $x\in\Zd$ we define any sausage for $(0,x)$ to be a zero step sausage-walk from the origin to $x$.

* We call a sausage-walk non-backtracking, if $\bar b^\omega_{i+1}\nin S^\omega_{i},\underline b^\omega_{i}\nin S^\omega_{i}$ and $\bar b^\omega_i\neq \underline  b^\omega_{i-1}$ for all $i$.

* We define $\Wcal^\ssss[A](x)$ as the set of all sausage-walks from the origin $0$ to $x$ and $\Wcal^{\ssss[A],\iota}(x)$ to be the set of all sausage-walks $\omega$ from $0$ to $x$ such that $\ve[\iota]\nin S^\omega_0$ and $\bar b_1^\omega\neq \ve[\iota]$.

Similarly to Definition <ref>, a key point in Definition <ref> is that a sausage-walk does not necessarily lead to a LA. For a rib-/sausage-walk $\omega$, we define

\begin{eqnarray}
\lbeq{defKanimal}
K[a,b](\omega)=\prod_{s=a}^{b-1}\prod_{s=t+1}^{b} \left(1 +\Ucal_{s,t}(\omega)\right),
\end{eqnarray}

where $-\Ucal_{s,t}(\omega)$ is the indicator that the ribs/sausages $S_s^\omega$ and $S_t^\omega$ intersect at some point in $\Zd$. Then, $K[0,|\omega|]$ is the indicator that all ribs/sausages of the walk are self-avoiding. Thus, if $K[0,|\omega|](\omega)=1$, then the union of the oriented bonds and all ribs/sausages of $\omega$ is a disjoint union and the resulting object is a LT/LA. The pair interaction in defKanimal thus gives a convenient description of when rib-/sausage-walks lead to a LT/LA, and this representation allows us to expand out this pair interaction in a convenient way.

To capture the contribution of the ribs/sausages we define

\begin{eqnarray}
\lbeq{defZanimal}
Z[a,b](\omega):=\prod_{i=a}^b z^{|S^{\omega}_i|},
\end{eqnarray}

and remark that $Z[a,b](\omega)=Z[a,c](\omega)Z[c+1,b](\omega)$ for $c\in[a,b)$. We often drop the argument $\omega$ for $\Ucal_{st}, K[a,b]$ and $Z[a,b]$ when this can cause no confusion. Further, we drop the superscript $A$ and $T$ for $\Wcal$ when we consider both models simultaneously. We can now write the two-point function as

\begin{eqnarray}
\lbeq{expanimalGen}
\bar G_{z}(x)&=&\sum_{n=0}^{\infty } t_{n}(x)z^n=\sum_{\omega\in\Wcal(x)}z^{|\omega|}Z[0,|\omega|]K[0,|\omega|].
\end{eqnarray}

In the NoBLE, we use an adaptation of the two-point function given by

\begin{eqnarray}
\lbeq{defGzIotaanimal}
\bar G^{\kappa}_z(x)&=&\sum_{\omega\in\Wcal^\kappa(x)}z^{|\omega|} Z[0,|\omega|]K[0,|\omega|].
\end{eqnarray}

We expand $\bar G^{\kappa}$ using the same set of graphs and laces as used by Hara and Slade in [18]:

Let $a,b\in \Nbold$ with $a<b$. For $s,t\in [a,b]\cap\Nbold$ with $s<t$, the edge between $s$ and $t$ is the tuple $(s,t)$. We abbreviate $st$ to denote $(s,t)$. We call a set of edges a graph. We call a graph connected, if for all $c\in[a,b]$ there exists an edge $st\in\Gamma$ such that $s\leq c\leq t$.
Let $\Bcal[a,b]$ be the set of all graphs on $[a,b]$ and $\Gcal[a,b]$ the set of all connected graphs on $[a,b]$.

We call a graph minimally connected or a lace if the removal of any edge would disconnect the graph and define $\Lcal[a,b]$ to be the set of all minimally connected graphs on $[a,b]$. We define the function ${\rm L}\colon \Gcal[a,b] \mapsto \Lcal[a,b]$ in a constructive manner as follows: For $\Gamma\in\Gcal[a,b]$, we let

\begin{eqnarray*}
&s_1=a, &t_1=\max \{t: at\in\Gamma\},  \\
&t_i=\max \{t\colon \exists s \leq t_{i-1}\text{ such that } st\in\Gamma\},  &s_i=\min \{s\colon  st_i\in\Gamma\}.
\end{eqnarray*}

This procedure ends after a finite number of steps $N$. We denote the resulting lace $L=\{s_1t_1,s_2t_2,\dots,s_Nt_N\}$ by ${\rm L}(\Gamma)$.
We call an edge $st\not\in L$ compatible to a lace $L$ if ${\rm L}(L\cup \{st\})=L$. We denote by $\Ccal(L)$ the set of all edges that are compatible with $L$.

We define, for $a>b$,

\begin{eqnarray}
\lbeq{defJanimal}
J[a,b]&=&\sum_{\Gamma\in\Gcal[a,b]}\prod_{st\in \Gamma}\Ucal_{st}=\sum_{L\in\Lcal[a,b]}\prod_{st\in L}\Ucal_{st}\prod_{s't'\in \Ccal(L)}(1+\Ucal_{s't'}).
\end{eqnarray}

Further, we see that

\begin{eqnarray}
K[a,b]&=&\prod_{t=a}^{b-1}\prod_{s=t+1}^{b} \left(1 +\Ucal_{s,t}(\omega)\right)=\sum_{\Gamma\in\Bcal[a,b]}\prod_{st\in \Gamma}\Ucal_{st}
\end{eqnarray}

and $K[a,a]=J[a,a]=1$. The key observation in the lace expansion is that, for $a<b$, we can write

\begin{eqnarray}
\lbeq{Krecrelation}
\end{eqnarray}

see e.g. <cit.>. We apply Krecrelation to expanimalGen with $a=0$ and $b=|\omega|>0$ to obtain

\begin{eqnarray}
\lbeq{Genanimal1}
\bar G_{z}(x)&=& \sum_{\omega\in\Wcal(x)} z^{|\omega|} Z[0,|\omega|] \sum_{i=0}^{|\omega|-1}J[0,i]K[i+1,|\omega|]\\
\lbeq{Genanimal2}
&&+\sum_{\omega\in\Wcal(x)} z^{|\omega|} Z[0,|\omega|]  J[0,|\omega|].
\end{eqnarray}

Here, the second term also contains the case where $|\omega|=0$.
We define the contribution of Genanimal2 to be $\bar \Xi_z(x)$, i.e.,

\begin{eqnarray}
\lbeq{defXianimal}
\bar \Xi_z(x)&=&\sum_{\omega\in\Wcal(x)} z^{|\omega|} Z[0,|\omega|]  J[0,|\omega|].
\end{eqnarray}

To further rewrite Genanimal1, we cut the non-backtracking rib-/sausage-walk at the $i$th bond of the backbone, $b_i=(y,y-\ve[\kappa])$, into a walk $\omega^1\in\Wcal$ from $0$ to $y$ (which could correspond to a one-point function for $|\omega^1|=0$), and a second walk $\omega^2\in\bigcup_{\kappa}\Wcal^{\kappa}$ from $y-\ve[\kappa]$ to $x$. This leads to

\begin{align}
\sum_{\omega\in\Wcal(x)} z^{|\omega|} Z[0,|\omega|] \sum_{i=0}^{|\omega|-1}J[0,i]K[i+1,|\omega|]
=& \sum_{y,\kappa}\sum_{\omega^1\in\Wcal(y)} z^{|\omega^1|+1} Z[0,|\omega^1|] J[0,|\omega^1|] \indic{y-\ve[\kappa]\nin S_{|\omega^1|}^{\omega^1}}\nnb
&\qquad \times\sum_{\omega^2\in\Wcal^{\kappa}(x-y+\ve[\kappa])}z^{|\omega^2|} Z[0,|\omega^2|]K[0,|\omega^2|]\nnb
=& z\sum_{y,\kappa}\bar \Psi^{\kappa}(y) \bar G_z^{\kappa}(x-y+\ve[\kappa]),
\lbeq{GenanimalSplit}
\end{align}

\begin{eqnarray}
\lbeq{defGzIotaanimal2}
\bar G_z^{\kappa}(x)&=&\sum_{\omega\in\Wcal^{\kappa}(x)} z^{|\omega|} Z[0,|\omega|]K[0,|\omega|],\\
\bar \Psi^{\kappa}_z(x)&=&\sum_{\omega\in\Wcal(y)} z^{|\omega|} Z[0,|\omega|] J[0,|\omega|] \indic{x-\ve[\kappa]\nin S_{|\omega|}^{\omega}}.
\end{eqnarray}

In this way, we have obtained a recurrence relation for the two-point function given by

\begin{eqnarray}
\lbeq{animalbasic1}
\bar G_z(x)&=&\bar \Xi_z(x)+z\sum_{y,\kappa}\bar \Psi^{\kappa}(y) \bar G_z^{\kappa}(x-y+\ve[\kappa]),
\end{eqnarray}

which is the first step towards Gx-in-ex-1. For Gx-in-ex-2, we instead consider

\begin{eqnarray}
\bar G_{z}(x)- \bar G^{\iota}_{z}(x)&=&\sum_{\omega\in\Wcal(x)\setminus \Wcal^{\iota}(x)} z^{|\omega|} Z[0,|\omega|]K[0,|\omega|].
\end{eqnarray}

As $\omega\in\Wcal(x)\setminus \Wcal^{\iota}(x)$ we know that $\ve[\iota]\in S_0^\omega$ or $\bar b_1=\ve[\iota]$.
For convenience, we define

\begin{eqnarray}
\lbeq{def-Eiota-forLTLA}
\1_{\iota}(\omega)=\indic{|\omega|>0}\indic{\bar b_1=\ve[\iota]}+\indic{\ve[\iota]\in S_0^\omega},
\end{eqnarray}

and remark that the non-backtracking condition of the rib-/sausage-walk excludes that $\ve[\iota]\in S_0^\omega$ and $\bar b_1=\ve[\iota]$ occur for the same walk. Therefore,

\begin{align}
\bar G_{z}(x)- \bar G^{\iota}_{z}(x)=&\sum_{\omega\in\Wcal(x)} z^{|\omega|} Z[0,|\omega|]K[0,|\omega|]\1_{\iota}(\omega)\nnb
\stackrel{\refeq{Krecrelation}}=&\sum_{\omega\in\Wcal(x)} z^{|\omega|} Z[0,|\omega|]\1_{\iota}(\omega)
\lbeq{ExpLTLATmp1}
\left(\sum_{i=0}^{|\omega|-1}J[0,i]K[i+1,|\omega|]+J[0,|\omega|]\right).
\end{align}

The contribution of $J[0,|\omega|]$ gives rise to

\begin{align}
\lbeq{defXiIotaanimal}
\bar \Xi^{\iota}_z(x)&=\sum_{\omega\in\Wcal(x)} \1_{\iota}(\omega) z^{|\omega|} Z[0,|\omega|]J[0,|\omega|].
\end{align}

Again, this term incorporates the contribution when $|\omega|=0$.

The dominant contribution of $J[0,i]K[i+1,|\omega|]$ in ExpLTLATmp1 is given by $|\omega|\geq 1, i=0$ and $b^{\omega}_0=(0,\ve[\iota])$, for which we see that

\begin{eqnarray*}
\lbeq{defGzIotaanimal-rewrite}
\sum_{\omega\in\Wcal(x)} \indic{b^{\omega}_1=(0,\ve[\iota])}z^{|\omega|} Z_R[0,|\omega|] J[0,0]K[1,|\omega|]=z\gj \bar G^{-\iota}_z(x-\ve[\iota]),
\end{eqnarray*}

where $\gj=\bar G^{\iota}_z(0)$. We extract this contribution explicitly, and split the remainder at $b_i=(y,y-\ve[\kappa])$, as in GenanimalSplit, which leads to

\begin{align}
\lbeq{ExpLTLATmp2}
\sum_{\omega\in\Wcal(x)}& z^{|\omega|} Z[0,|\omega|]\sum_{i=0}^{|\omega|-1}J[0,i]K[i+1,|\omega|]\1_{\iota}(\omega)(1-\delta_{i,0}\indic{b_1^\omega=(0,\ve[\iota])}) \\
z^{|\omega^1|+|\omega^2|+1} Z[0,|\omega^1|](\omega^1) Z[0,|\omega^2|](\omega^2) \nnb
&\qquad\times J[0,|\omega^1|](\omega^1)K[0,|\omega^2|](\omega^2) (
\1_{\iota}(\omega^1)+\indic{|\omega^1|=0} \indic{y\neq 0} \indic{y-\ve[\kappa]=\ve[\iota]})\indic{y-\ve[\kappa]\nin S^{\omega^1}_{|\omega^1|}}.\nn
\end{align}

We define

\begin{align}
\lbeq{defPiIotaanimalIndic}
\1^{\Pi}_{\iota}(\omega,x,\kappa)&=(\1_{\iota}(\omega)+ \indic{|\omega|=0}\indic{x\neq 0}\indic{x-\ve[\kappa]=\ve[\iota]}),\\
\Pi^{\iota,\kappa}_z(x)&=\sum_{\omega\in\Wcal(x)}\1^{\Pi}_{\iota}(\omega,x,\kappa)
\indic{x-\ve[\kappa]\nin S_{|\omega|}^{\omega}} z^{|\omega|+1} Z[0,|\omega|]J[0,|\omega|],
\lbeq{defPiIotaanimal}
\end{align}

and see that sums over $\omega^1$ and $\omega^2$ in defGzIotaanimal-rewrite factorize, to conclude that

\begin{eqnarray}
\lbeq{animalbasic2}
\bar G_{z}(x)- \bar G^{\iota}_{z}(x) &=&\bar \Xi^{\iota}_z(x)+z\gj \bar G^{-\iota}_z(x-\ve[\iota])+\sum_{y,\kappa}\Pi_z^{\iota,\kappa}(y)\bar G_{z}^{\kappa}(x-y+\ve[\kappa]).
\end{eqnarray}

This completes the derivation of the expansion for LT and LA for $\bar G_{z}(x)$. To obtain Gx-in-ex-1 and Gx-in-ex-2, we need to divide animalbasic1 and animalbasic2 by $g_z$, as we will explain in more detail in the next section.

§.§ Definitions used in the generalized analysis

In this section we complete the expansion as stated in Proposition <ref> and used in the general analysis of [12].
For the analysis, we use the normalized two-point functions LT and LA defined as

\begin{align}
G_z(x)&=\frac 1 {g_z} \bar G_z(x),\qquad\qquad G^\iota_z(x)=\frac 1 {g_z} \bar G^\iota_z(x).
\end{align}

This has the advantage that the analysis and bounds on the lace-expansion coefficients are naturally divided into two parts.
The one-point functions $g_z$ and $g_z^\iota$ and their influence, see conj-gzc, are controlled using the bootstrap function $f_1$, see defFunc1.
The spatial dependence of the two-point functions is controlled using the bootstrap function $f_2$, see defFunc2.

To improve the performance of our analysis,

we extract some dominant contributions of the lace-expansion coefficients to be used explicitly within the analysis, so as to improve the numerical accuracy of the method. These explicit terms are defined in Section <ref>. Before that we now complete the proof of Proposition <ref> by identifying the lace-expansion coefficients arising in it.

The NoBLE coefficients can be written into an alternating series of non-negative functions. The series arise in a natural way by the negative signs of the ${\mathcal U}_{st}$ terms in the expansion for LTs and LAs, see defJanimal as well as <cit.>, as we now explain in more detail:

For $n\geq 1$ and $N\geq 1$,
let $\Lcal^{\ssc[N]}[0,n]\subseteq\Lcal[0,n]$ be the set of all laces $L\subseteq\Lcal[0,n]$ that consist of exactly $N$ edges.

\begin{align}
\lbeq{defJNanimal}
J^{\ssc[N]}[a,b](\omega)&= \sum_{L\in\Lcal^\ssc[N][a,b]}\prod_{st\in L}\Ucal_{st}\prod_{s't'\in \Ccal(L)}(1+\Ucal_{s't'}).
\end{align}

A sausage/rib-walk $\omega$ for which the indicator $J^\ssc[N]$ equals one has $N$ intersecting sausages/ribs. These intersections are characterized by the lace $L$ in defJNanimal.

This gives a convenient interpretation to the NoBLE coefficients that allows for sharp bounds. For $N\geq 0$ and $x\in\Zd$, we define

\begin{align}
\lbeq{defXiNanimal}
\bar \Xi^{\ssc[N]}_z(x) &= (-1)^N \sum_{\omega\in\Wcal(x)} z^{|\omega|} Z[0,|\omega|]  J^\ssc[N][0,|\omega|],\\
\lbeq{defXiNIotaanimal}
\bar \Xi^{\ssc[N],\iota}_z(x)&=(-1)^N \sum_{\omega\in\Wcal(x)} z^{|\omega|} Z[0,|\omega|]J^\ssc[N][0,|\omega|]\1_{\iota}(\omega),\\
\lbeq{defPsiNanimal}
\bar \Psi^{\ssc[N],\kappa}_z(x)&=(-1)^N\sum_{\omega\in\Wcal(y)} z^{|\omega|} Z[0,|\omega|] J^\ssc[N][0,|\omega|] \indic{x-\ve[\kappa]\nin S^{\omega}_{|\omega|}},\\
\Pi^{\ssc[N],\iota,\kappa}_z(x)&=(-1)^N \sum_{\omega\in\Wcal(x)} z^{|\omega|+1} Z[0,|\omega|] J^\ssc[N][0,|\omega|]\indic{x-\ve[\kappa]\nin S_{|\omega|}^{\omega}}
\1^{\Pi}_{\iota}(\omega,x,\kappa),
\lbeq{defPiNanimal}
\end{align}

to be the restrictions of the NoBLE coefficients to laces of fixed size, with the convention that $J^\ssc[0][0,|\omega|]=\delta_{0,|\omega|}$.

The dominant contributions of these coefficients are given by

\begin{align}
\lbeq{dominate-piece-part1}
\bar \Xi^{\ssc[0]}_z(0) =&\bar G_z(0)=g_z,\quad \bar \Psi^{\ssc[0],\kappa}_z(0)= \bar G^{\iota}_z(0)=\gj,\quad
\Xi^{\ssc[0],\iota}_z(0)=\bar G_z(\ve[\iota]).
%\\ \lbeq{dominate-piece-part2}
%\Pi^{\ssc[0],\iota,\kappa}_z(0)=&\sum_{A\ni 0}z^{|A|}\indic{\ve[\iota]\in A}\indic{-\ve[\kappa]\nin A}\leq z (1-\delta_{\iota,-\kappa})\bar G^{-\kappa}_z(\ve[\iota]).
\end{align}

We use these terms explicitly in our analysis. We perform the NoBLE analysis using the following coefficients:

\begin{align}
\lbeq{defXiNanimalPrime}
\Xi^{\ssc[N]}_z(x)&=\frac 1 {g_z} \big(\bar \Xi^{\ssc[N]}_z(x)-\delta_{0,x}\delta_{0,N}\bar \Xi^{\ssc[0]}_z(0)\big),\qquad \Xi^{\ssc[N],\iota}_z(x)=\frac 1 {g_z} \bar \Xi^{\ssc[N],\iota}_z(x),\\
\lbeq{defPsiNanimalPrime}
\Psi^{\ssc[N],\kappa}_z(0)&=\frac 1 {g^\iota_z} \big(\bar \Psi^{\ssc[N],\kappa}_z(x)-\delta_{0,x}\delta_{0,N}\bar \Psi^{\ssc[0],\kappa}_z(0)\big),\\
\Xi_z(x)&=\sum_{N=0}^{\infty}(-1)^N \Xi^{\ssc[N]}_z(x),\qquad \Xi^{\iota}_z(x)=\sum_{N=0}^{\infty}(-1)^N \Xi^{\ssc[N],\iota}_z(x),\\
\Psi^{\kappa}_z(x)&=\sum_{N=0}^{\infty}(-1)^N \Psi^{\ssc[N],\kappa}_z(x),\qquad \Pi^{\iota,\kappa}_z(x)=\sum_{N=0}^{\infty}(-1)^N \Pi^{\ssc[N],\iota,\kappa}_z(x),
\lbeq{LALTCoefficientNormalization}
\end{align}

where we explicitly subtract the dominant contributions, of which most of them are also present for the NBW. The situation simplifies considerably for LTs, due to the absence of double connections, so that

Ξ^[0],ι_z(x)=δ_0,x G_z([ι]),

while, using def-Eiota-forLTLA, as well as defPiIotaanimalIndic–defPiIotaanimal,

Π^[0],ι,κ_z(x)=δ_0,x z ∑_T0∈T z^|T|x-[κ]T [ι]∈T.

For LAs, several further contributions due to $x\neq 0$, which require a double connection, arise.
Using the above notation we obtain Gx-in-ex-1-Gx-in-ex-2 by dividing the equations animalbasic1 and animalbasic2 by $g_z$. Thereby, we have completed the proof of Proposition <ref> and have identified the NoBLE coefficients appearing in it.

§.§ Split for the coefficients

In this section, we define further terms that we extract from the NoBLE coefficients to be used in the analysis of [12]. By extracting these terms we reduce the size of the perturbation, which ultimately improves the numerical accuracy of our analysis. We give first the definition for LAs and then explain how terms simplify for LTs.

Definition of the explicit terms for $N=0$.
For a LA $A$ and $x,y\in\Zd$, we denote by $x\dbct{A} y$ the event that $x$ and $y$ are doubly connected via bonds in $A$. By $d_A(x,y)$ we denote the intrinsic distance of $x,y$ in $A$, meaning the length of the shortest connection of $x$ and $y$ using bonds in $A$. For $N=0$, we define

\begin{align}
\lbeq{LA-Split-Def-1}
\Xi^{\ssc[0]}_{\alpha,z}(x)&=(1-\delta_{0,x})\frac {1}{g_z}\sum_{A\colon 0,x\in A}\indic{0\dbct{A} x}\indic{d_A(0,x)=1}z^{|A|}, \\
\Psi^{\ssc[0],\kappa}_{\alpha,{\sss I},z}(x)&=(1-\delta_{0,x})\frac {1}{g^\iota_z}\sum_{A\colon 0,x\in A}\indic{0\dbct{A} x}
\indic{x-\ve[\kappa]\nin A}\\
&\qquad \times (\indic{d_A(0,x)=1}\indic{x=\ve[\kappa]}+\indic{|\ve[\kappa]-x|=1}\indic{d_A(0,x)=2}),\nn\\
\Psi^{\ssc[0],\kappa}_{\alpha,{\sss II},z}(x)&=(1-\delta_{0,x})\frac {1}{g^\iota_z}\sum_{A\colon 0,x\in A}\indic{0\dbct{A} x}\indic{d_A(0,x)=1} \indic{x-\ve[\kappa]\nin A},
\end{align}

\begin{align}
\lbeq{LA-Split-Def-XiIotaAlphaa}
\Xi^{\ssc[0],\iota}_{\alpha,{\sss I},z}(x)&=\frac {1}{g_z}
\sum_{A\colon 0,x,\ve[\iota]\in A} z^{|A|}\indic{0\dbct{A} x} \left(\delta_{0,x}+\delta_{\ve[\iota],x} +(1-\delta_{0,x})\indic{d_A(\ve[\iota],x)=1}\right),\\
\lbeq{LA-Split-Def-XiIotaAlphab}
\Xi^{\ssc[0],\iota}_{\alpha,{\sss II},z}(x)&=\frac {1}{g_z}\sum_{A\colon 0,x,\ve[\iota]\in A} z^{|A|}\indic{0\dbct{A} x} \left(\delta_{0,x}+\delta_{\ve[\iota],x}
\Pi^{\ssc[0],\iota,\kappa}_{\alpha,z}(x)&=
\sum_{A\colon 0,x,\ve[\iota]\in A} z^{|A|+1}\indic{0\dbct{A} x}\indic{x-\ve[\kappa]\nin A}\left(\delta_{\ve[\iota],x}+\delta_{\ve[\iota]+\ve[\kappa],x}\indic{d_A(\ve[\iota]+\ve[\kappa],\ve[\iota])=1}\right).
\lbeq{LA-Split-Def-1a}
\end{align}

For $N=1$, the terms are more involved and are given by

\begin{align}
\lbeq{LA-Split-Def-None}
\Xi^{\ssc[1]}_{\alpha,z}(x)&=\frac 1 {g_z}\sum_{\omega\in\Wcal(x)} z^{|\omega|} Z[0,|\omega|]J^{\ssc[1]}[0,|\omega|]\indic{\bb^\omega_0=0}\\
&\qquad\times \left(\delta_{0,x}+\delta_{|\omega|,1}\delta_{\tb^\omega_0,x}
+\indic{x\in S_0\cap S_{|\omega|}}\indic{d_{S_0}(0,x)=1} +
\indic{0\in S_0\cap S_{|\omega|}\not \ni x}\indic{d_{S_{|\omega|}(0,x)=1}}\right),\nnb
\Psi^{\ssc[1],\kappa}_{\alpha,{\sss II},z}(x)&=\frac 1 {g^\iota_z}\sum_{\omega\in\Wcal(x)} z^{|\omega|} Z[0,|\omega|]J^{\ssc[1]}[0,|\omega|]\indic{\bb^\omega_0=0}
\indic{x-\ve[\kappa]\nin S_{|\omega|}^{\omega}}\\
&\qquad \times\left(\delta_{0,x}+\delta_{|\omega|,1}\delta_{\tb^\omega_0,x}+\indic{x\in S_0\cap S_{|\omega|}}\indic{d_{S_0}(0,x)=1} +
\indic{0\in S_0\cap S_{|\omega|}\not \ni x}\indic{d_{S_{|\omega|}(0,x)=1}}\right),\nnb
\Psi^{\ssc[1],\kappa}_{\alpha,{\sss I},z}(x)&=(\delta_{0,x}+\delta_{x,\ve[\kappa]})
\Psi^{\ssc[1],\kappa}_{\alpha,{\sss II},z}(x)\lbeq{LA-Split-Def-10}\\
&\quad +\frac {2d D(x-\ve[\kappa])}{g^\iota_z}
\sum_{\omega\in\Wcal(x)} z^{|\omega|} Z[0,|\omega|]J^{\ssc[1]}[0,|\omega|]\indic{x-\ve[\kappa]\nin S_{|\omega|}^{\omega}}\delta_{|\omega|,2}
\indic{\bb^\omega_0=0}\delta_{\tb^\omega_0,\bb^\omega_1}\delta_{\tb^\omega_1,x}.\nn
\end{align}

All sausage-walks contributing to these sums have the following in common: (a) The first step of the backbone $\bb^\omega_0$ starts at the origin.
(b) The sausages $S_0$ and $S_{|\omega|}$ intersect either at $0$ or at $x$. (c) At least one bond is explicitly present. Most of the times this bond is $(0,x)$, forced by $d_A(0,x)=1$.

We define the remainder terms arising through this split by

\begin{align}
\Xi^{\ssc[0]}_{{\sss R},z}&=\Xi^{\ssc[0]}_{z}-\Xi^{\ssc[0]}_{\alpha,z},\qquad\qquad\quad
\Xi^{\ssc[1]}_{{\sss R},z}=\Xi^{\ssc[1]}_{z}-\Xi^{\ssc[1]}_{\alpha,z},\\
\Psi^{\ssc[0],\iota}_{{\sss R, I},z}&=\Psi^{\ssc[0],\iota}_{z}-\Psi^{\ssc[0],\iota}_{\alpha,{\sss I},z},\qquad\qquad
\Psi^{\ssc[0],\iota}_{{\sss R, II},z}=\Psi^{\ssc[0],\iota}_{z}-\Psi^{\ssc[0],\iota}_{\alpha,{\sss II},z},\\
\Psi^{\ssc[1],\iota}_{{\sss R, I},z}&=\Psi^{\ssc[1],\iota}_{z}-\Psi^{\ssc[1],\iota}_{\alpha,{\sss I},z},\qquad\qquad\
\Psi^{\ssc[1],\iota}_{{\sss R, II},z}=\Psi^{\ssc[1],\iota}_{z}-\Psi^{\ssc[1],\iota}_{\alpha,{\sss II},z},\\
\Xi^{\ssc[0],\iota}_{{\sss R, I},z}&=\Xi^{\ssc[0],\iota}_{z}-\Xi^{\ssc[0],\iota}_{\alpha,{\sss I},z},\qquad\qquad\ \
\Xi^{\ssc[0],\iota}_{{\sss R, II},z}=\Xi^{\ssc[0],\iota}_{z}-\Xi^{\ssc[0],\iota}_{\alpha,{\sss II},z},\\
\Pi^{\ssc[0],\iota,\kappa}_{{\sss R},z}&=\Pi^{\ssc[0],\iota,\kappa}_{z}-\Pi^{\ssc[0],\iota,\kappa}_{\alpha,z}.
\end{align}

This completes the necessary split for LAs.

Lattice trees.
We use the same definitions for LTs, where we sum over LTs $T$ instead of LAs $A$.
However, for LTs the terms simplify considerably, as double connections are not possible. This is especially true for $N=0$, for which, for all $x\in\Zd$ (recall split-LTs-N0),

\begin{align}
\lbeq{split-LTs-N0a}
\Xi^{\ssc[0]}_{\alpha,z}(x)=\Psi^{\ssc[0],\iota}_{\alpha,{\sss I},z}(x)=\Psi^{\ssc[0],\iota}_{\alpha,{\sss II},z}(x)=0.
\end{align}

Further, the split actually captures the complete contribution of $\Xi^{\ssc[0],\iota}_z$, since

\begin{align}
\lbeq{split-LTs-N0b}
\Xi^{\ssc[0],\iota}_{z}(x)=\Xi^{\ssc[0],\iota}_{\alpha,{\sss I},z}(x)&=\Xi^{\ssc[0],\iota}_{\alpha,{\sss II},z}(x)=\delta_{0,x}  \frac {\bar G_z(\ve[\iota])}{g_z}=\delta_{0,x} G_z(\ve[\iota]).
\end{align}

This completes the derivation of the split of the coefficients as used in <cit.>.

§.§ Assumptions on the model

In this section, we verify most of the assumptions necessary to apply the analysis of [12].
We start by proving the assumption that are independent of the NoBLE: <cit.>.

§.§.§ Assumption on the two-point function

We begin with <cit.> as it will help us prove <cit.>.

<cit.>:  For $z\in[0,z_c)$, the functions $ z\mapsto\aabz$ and $z\mapsto\aaz$ are continuous.

To verify this assumption, we choose $\aabz=zg_z$ and $\aaz=zg_z^\iota$, with

\begin{align}
\gj=\sum_{A\colon A\ni 0}z^{|A|} \qquad \text{ and }\qquad \gj=\sum_{A\colon \ve[1]\nin A\ni 0}z^{|A|}.
\end{align}

By Abel's Theorem, the one-point function is continuous within the radius of convergence of these power series, which is at most $z_c$, see defLTLAsusceptibility and the text thereafter. Thus, also $z\mapsto\aabz$ and $z\mapsto\aaz$ are continuous.

<cit.>:There exists a $z_I\in[0,z_c)$ such that

\begin{align}
\lbeq{assInitialXBound}
G_{z}(x)\leq B_{1/(2d-1)}(x)=\frac {2d-2}{2d-1} C_{1/2d}(x)
\end{align}

for all $x\in \Zd$ and $z\in[0,z_I]$.

To prove this statement for LTs and LAs we adapt an argument used in <cit.>.
We know that each LT/LA containing $0$ and $x$ contains a path of bonds that connects $0$ and $x$. Each point of the path is connected to at most one rib/sausage.
The weight of all possible rib/sausages $z^{|S_i|}$ (see defZanimal) can be bounded by $g_z$, as each rib/sausage $S_i$ is itself a LT/LA.
For $x\neq 0$ we can improve this by bounding the weight by $\gj$ instead. This is possible as each rib/sausage needs to avoid at least the next/previous step of the path from $0$ to $x$. From this, we conclude for $x\neq 0$ that

\begin{align}
\lbeq{LTInitalBound-goal}
\bar G_z(x)\leq \gj \sum_{\omega\in\Wcal^{\ssss[NBW]}(x)} (z\gj)^{|\omega|}=\gj B_{z\gj}(x),
\end{align}

where $\Wcal^{\ssss[NBW]}(x)$ is the set of all NBWs (see the text above NBWGen) starting at $0$ and ending at $x$, and $|\omega|$ is the number of steps of the NBW $\omega$. The inequality LTInitalBound-goal then follows for all $z$ for which $z\gj\leq (2d-1)^{-1}$ and $x\neq 0$. We define

\begin{align}
\lbeq{DefinitionOfzI}
z_I:=\sup\Big\{z\colon zg_{z}^\iota\leq \frac 1 {2d-1} \Big\}.
\end{align}

This is well defined as $\aaz=zg_{z}^\iota$ is continuous and non-decreasing in $z$ with $zg_{z}^\iota=0$ when $z=0$. To complete the proof of <cit.>, we still need to prove that $z_I<z_c$.

For this we note that $g_z-g_z^\iota=\bar G_z(\ve[1])$ and use LTInitalBound-goal to obtain

\begin{align}
\lbeq{LTInitalBound-goal-step}
\bar \chi(z)=\sum_x\bar G_z(x)\leq \bar G_z(0)+\sum_{x\neq 0} \gj B_{z\gj}(x)\leq \gj \big(1+ B_{z\gj}(\ve[1]) + \chi^{\sss \rm NBW}(z\gj)\big).
\end{align}

From this we conclude that $z_I\leq z_c$ as otherwise $\chi(z_c)=\infty$, while $\chi^{\sss \rm NBW}(z_cg_{z_c}^\iota)$ on the right-hand side is finite.
To exclude that $z_I=z_c$, we see that LTInitalBound-goal also holds when we replace $\gj$ with

\begin{align}
\lbeq{gz'-def}
g'_z=\min_{\kappa\neq 1} \sum_{A:\ve[1],\ve[\kappa]\not \in A\ni 0}z^{|A|}
\end{align}

for all ribs/sausages except those at $0$ and $x$. We define $z'$ as the value $z$ such that $z'g'_{z'}=(2d-1)^{-1}$, and  conclude as in LTInitalBound-goal-step that $z_c \geq z'>z_I$.
Recalling $\bar G_z(x)=g_z G_z(x)$ and $\gj<g_z$ we  conclude LTInitalBound-goal from assInitialXBound. This concludes the proof of <cit.>.

Additionally, we prove the lower bound $z_I\geq (2d-1)^{-1}\e^{-1}$. As this uses  ideas not used elsewhere, we move the proof to Appendix <ref> (see Lemma <ref>).

<cit.>: Growth of the two-point function.
We need to show that for every $x\in \Zd$, the two-point functions $z\mapsto G_z(x)$ and $z\mapsto G^\iota_z(x)$
are non-decreasing and differentiable in $z\in(0,z_c)$. Further, we need to show that for all $\varepsilon>0$, there exists a constant $c_{\varepsilon}\geq 0$ such that for all $z\in(0,z_c-\varepsilon)$ and $x\in\Zd\setminus\{0\}$,

\begin{eqnarray}
\lbeq{assGzDiffBound}
\frac d {dz} G_z(x)\leq c_{\varepsilon} (G_z\star D\star G_z)(x),
\quad \text{ and therefore }\quad \frac d {dz} \hat G_z(0)\leq c_{\varepsilon} \hat G_z(0)^2.
\end{eqnarray}

Finally, we need to show that for each $z\in(0,z_c)$, there exists a constant $K(z)<\infty$ such that $\sum_{x\in\Zd} |x|^2 G_{z}(x)<K(z)$. We will do this now.

As a generating function of a non-negative sequence (see defLTLATwoPoint), the two-point function is clearly non-decreasing in the parameter $z$
as well as differentiable in $z$ for $z\in(0,z_c)$. Next, we first prove the bound on the derivative in assGzDiffBound for LTs.
We know that a LT $T$ with $|T|$ edges contains $|T|+1$ vertices. We use this property to compute for $x\neq 0$

\begin{align}
\lbeq{LTDiffBound-tmp1}
\sum_{T\ni 0,x} \frac d {dz} z^{|T|}=\sum_{T\ni 0,x} {|T|} z^{|T|-1}=\sum_{y\neq 0}\sum_{T\ni 0,x,y} z^{|T|-1}.
\end{align}

As a LT $T\ni x,y$ contains no loops, the path from $x$ to $y$ is unique. We denote this path by $b^T(x,y)$.
By $u$ we denote the last vertex that the paths from $0$ to $x$ and from $0$ to $y$ have in common.
For $u\neq x$ we split the walk at $u$ into three individual trees and bound the contributions of these individual trees by two-point functions.
Doing this, we have to take into account that in LTDiffBound-tmp1 the tree $T$ is only weighted by $z^{|T|-1}$,
so that we have to choose one bond of the tree that does not carry the weight $z$.
We choose the first step of the path from $u$ to $x$ to be this bond.
For $u=x$ there exists no first step. In this case, we choose the last step of the path from $0$ to $x$ to be the bond without weight $z$. Using that $\bar G_{z}(0)\geq 1$, we obtain the bound

\begin{align}
\sum_{T\ni 0,x} \frac d {dz} z^{|T|}\leq&\ \sum_{y\neq 0} \sum_{u\neq x,v}  \bar G^{(t)}_z(u)2d D(v-u)\bar G^{(t)}_{z}(x-v) \bar G^{(t)}_z(u-y)\lbeq{LTDiffBound-tmp2}\\
&\ + \sum_{y\neq 0} \sum_{v} \bar G^{(t)}_z(v)2dD(v-x) \bar G^{(t)}_{z}(x-y) \nnb
\leq &\ 4d (\bar G^{(t)}_z\star D\star \bar G^{(t)}_z)(x) \sum_y \bar G^{(t)}_z(y)= 4d \hat {\bar G}^{(t)}_z(0) (\bar G^{(t)}_z\star D\star \bar G^{(t)}_z)(x).
\nn
\end{align}

For the normalized two-point function, we conclude

\begin{align}
\frac d {dz} G^{(t)}_z(x)&= \frac d {dz} \frac {\bar G^{(t)}_z(x)}{g^{(t)}_z}=
\frac 1 {g^{(t)}_z}\sum_{T\ni 0,x} \frac d {dz} z^{|T|} -\frac {\bar G^{(t)}_z(x)} {(g^{(t)}_z)^2} \sum_{T\ni 0} \frac d {dz} z^{|T|} \nnb
&\leq 4d (g^{(t)}_z)^2 \hat G^{(t)}_z(0)(G^{(t)}_z\star D\star G^{(t)}_z)(x),
\lbeq{LTDiffBound-tmp3}
\end{align}

as required. To conclude such an inequality for LAs, we note that an animal with $|A|$ bonds contains at least $|A|/d$ vertices and as for the LT compute that

\begin{align}
\frac d {dz} G^{(a)}_z(x)&=\frac 1 {g^{(a)}_z}\sum_{A\ni 0,x} |A|z^{|A|-1}\leq \frac {d} {g^{(a)}_z}\sum_{A\ni 0,x,v} z^{|A|-1}\nnb
&\leq 4d^2 (g^{(a)}_z)^2 \hat G^{(a)}_z(0)(G^{(a)}_z\star D\star G^{(a)}_z)(x).
\lbeq{LTDiffBound-tmp4}
\end{align}

As the last step we prove that for all $z< z_c$ there exists $K(z)<\infty$ such that
$\sum_{x\in\Zd} |x|^2 G_{z}(x)<K(z)$. In <cit.>, it is proved that the connectivity constants $\lambda=1/z_c$ can be used to prove that

\begin{align}
t_n(0) \leq  \lambda^n (n+1).
\end{align}

Let $|x|_\infty:=\max_{i=1}^d |x_i|$ be the supremum norm and compute

\begin{align}
\bar G_z(x) = & \sum_{n=|x|_\infty}^\infty t_n(x)z^n\leq \sum_{n=|x|_\infty}^\infty t_n(0)z^n
\leq \sum_{n=|x|_\infty}^\infty (z/z_c)^n (n+1).
\end{align}

From this we conclude that $\bar G_z(x)$ decays exponentially, i.e., there exists $c,m(z)\in(0,\infty)$ such that

\begin{align}
\bar G_z(x) \leq&\sum_{n=|x|_\infty}^\infty (n+1)(z/z_c)^{n}\leq c \e^{-m(z) |x|_\infty}.
\end{align}

We use this bound to conclude that

\begin{align}
\sum_{x\in\Zd} |x|^2\bar G_z(x) \leq&\sum_{x\in\Zd} |x|^2 c \e^{-m(z) |x|_\infty}
\stackrel{|x|^2\leq d^2 |x|_\infty^2}\leq \sum_{n=1}^{\infty} (dn)^2c \e^{-m(z) n}\sum_{x\colon |x|_\infty=n}1\nnb
\leq&\ d^3 \sum_{n=1}^{\infty} n^{d+1}c \e^{-m(z) n}:=K(z)<\infty,
\end{align}

which proves the desired statement.

§.§.§ Assumptions on the NoBLE-coefficients

In this section, we verify the assumptions on the NoBLE coefficients formulated in <cit.>.

<cit.> Symmetry of the model.We denote by $\mathcal{P}_d$ the set of all permutations of $\{1,2,\dots, d\}$. For $\nu\in \mathcal{P}_d$, $\delta\in\{-1,1\}^d$ and $x\in\Zd$, we define $p(x;\nu,\delta)\in\Zd$ to be the vector with entries $(p(x;\nu,\delta))_j=\delta_j x_{\nu_j}$. We say that a function $f\colon \Zd\mapsto \Rbold$ is totally rotationally symmetric (TRS) when $f(x)=f(p(x;\nu,\delta))$ for all $\nu\in \mathcal{P}_d$ and $\delta\in\{-1,1\}^d$.

Total rotational symmetry is natural on $\Zd$, e.g. the two-point function $G_z$ as well as the NBW and SRW two-point functions have this symmetry.
We next argue that the NoBLE coefficients have similar symmetries.

<cit.>. Let $\iota,\kappa\in\{\pm 1,\pm 2,\dots,\pm d\}$. The following symmetries hold for all $x\in\Zd$, $z\leq z_c$, $N\in\Nbold$ and $\iota,\kappa$:

\begin{eqnarray}
\lbeq{Symmetrie-goal}
\Xi^\ssc[N]_z(x)&=& \Xi^\ssc[N]_z(-x), \qquad  \qquad \qquad
\Xi^{\ssc[N],\iota}_z(x)= \Xi^{\ssc[N],-\iota}_z (-x),\\
\lbeq{Symmetrie-goal2}\Psi^{\ssc[N],\iota}_z(x)&=& \Psi^{\ssc[N],-\iota}_z(-x), \qquad \qquad\
\Pi^{\ssc[N],\iota,\kappa}_z(x)= \Pi^{\ssc[N],-\iota,-\kappa}_z (-x).
\end{eqnarray}

For all $N\in\Nbold$, the coefficients

\begin{align}
\lbeq{TRS-sums}
\Xi^\ssc[N](x),\qquad \sum_{\iota}\Psi^{\ssc[N],\iota}_z(x),
\qquad \sum_{\iota}\Xi^{\ssc[N],\iota}_z(x) \quad \text{and}\quad \sum_{\iota,\kappa}\Pi^{\ssc[N],\iota,\kappa}_z(x),
\end{align}

as well as the remainder terms of the split

\begin{align}
\lbeq{TRS-sums2}\Xi^{\ssc[N]}_{{\sss R},z}(x),\quad \sum_{\iota}\Psi^{\ssc[N],\iota}_{{\sss R, I},z}(x),
\quad \sum_{\iota}\Psi^{\ssc[N],\iota}_{{\sss R, II},z}(x),
\quad \sum_{\iota}\Xi^{\ssc[0],\iota}_{{\sss R, I},z}(x)
\quad \sum_{\iota}\Xi^{\ssc[0],\iota}_{{\sss R, II},z}(x),
\quad \sum_{\iota,\kappa}\Pi^{\ssc[0],\iota,\kappa}_{{\sss R},z}(x),
\end{align}

are totally rotationally symmetric functions of $x\in\Zd$.
Further, the dimensions are exchangeable, i.e., for all $\iota,\kappa$,

\begin{align}
\lbeq{assCoefficentsDimInterchange}
\hat \Psi^{\ssc[N],\iota}_z(0)=\ \hat \Psi^{\ssc[N],\kappa}_z(0),\qquad
\hat \Xi^{\ssc[N],\iota}_z(0)=\ \hat \Xi^{\ssc[N],\kappa}_z(0),\qquad
\sum_{\kappa'}\hat \Pi^{\ssc[N],\iota,\kappa'}_z(0)=\sum_{\iota'}\hat \Pi^{\ssc[N],\iota',\kappa}_z(0).
\end{align}

By the definition of the NoBLE coefficients in Section <ref>, it is easy to see that Symmetrie-goal, Symmetrie-goal2 and assCoefficentsDimInterchange hold.
The TRS stated in TRS-sums, TRS-sums2 might be less obvious.
The definition of $\Xi^\ssc[N]_z$ does not include constraints on specific directions, so it is not difficult to see that $x\mapsto \Xi^\ssc[N]_z(x)$ is TRS for all $N\geq 0$.
The other three NoBLE coefficients are not TRS as their definition includes constraints on one or two specific directions.
For example, the coefficient $\Psi^{\ssc[N],\kappa}_z(x)$ includes the constraint that $x-\ve[\kappa]$ is not in the last rib/sausage.
When we sum over $\kappa$, though, the directional constraint is averaged out and thus $\sum_{\kappa}\Psi^{\ssc[N],\kappa}_z(x)$ is TRS.
For the same reason, the sums over $\iota$ and $\iota,\kappa$ in TRS-sums, as well as the stated remainder terms, are TRS.
The arguments given above hold, when the coefficients are well defined, which is definitely the case for all $z<z_c$.

<cit.> states that the symmetry properties also hold for $z=z_c$, where it is not even obvious that these objects are well defined.
We verify the left-continuity in <cit.> below, from which the symmetries will follow also for $z=z_c$.
Further, inspection of the proof in [12] shows that the symmetries are only used for $z<z_c$.

<cit.> Relation between coefficients.For all $x\in\Zd$, $z\leq z_c$, $N\in\Nbold$ and $\iota,\kappa\in\{\pm 1,\pm 2,\dots,\pm d\}$, the following bounds hold:

\begin{align}
\lbeq{XidominatespsiImproved}
\Psi^{\ssc[N],\kappa}_z(x)\leq&\frac {\aabz}{ \aaz} \Xi^\ssc[N]_z(x),
\qquad\qquad\Pi^{\ssc[N],\iota,\kappa}_z(x)\leq  \aabz \Xi^{\ssc[N],\iota}_z(x).
\end{align}

We first compare $\bar \Xi^{\ssc[N]}$ and $\bar \Psi^{\ssc[N],\kappa}$, in defXiNanimal,defPsiNanimal and see that they
differ by the additional condition $x-\ve[\kappa]\nin S^{\omega}_{|\omega|}$, so that $\bar \Xi^{\ssc[N]}(x)\leq \bar \Psi^{\ssc[N],\kappa}(x)$.
Considering the normalisation LALTCoefficientNormalization, we see that $\Xi^{\ssc[N]}$ was normalised using $g_z$, while $\Psi^{\ssc[N],\kappa}$ was normalised using $\gj$. As $\aabz/\aaz=g_z/\gj$ we conclude that the left inequality in XidominatespsiImproved holds.

Regarding $\Xi^{\ssc[N],\iota}$ and $\Pi^{\ssc[N],\iota,\kappa}$, defined in defXiNIotaanimal and defPiNanimal, note that $\Pi^{\ssc[N],\iota,\kappa}$ contains the extra factor $\1_{\iota}(\omega)=\1^{\Pi}_{\iota}(\omega,x,\kappa)$ for $N\geq 1$, as well as an additional factor $z$. Further, we do not normalize $\Pi^{\ssc[N],\iota,\kappa}$, while $\Xi^{\ssc[N],\iota}$ is normalized with a factor $1/g_z$, creating the stated factor $\aabz=zg_z$, which proves the bound for $N\geq 1$. To obtain the equation also for $N=0$, we review defPiIotaanimal and see that $\1_{\iota}(\omega)$ and $\1^{\Pi}_{\iota}(\omega,x,\kappa)$ only differ when $x\neq 0$, which is not possible for LTs, as double connections are not present. For LAs, we see that

\begin{align}
zg_z \Xi^{\ssc[0],\iota}(x)-\Pi^{\ssc[0],\iota,\kappa}(x)
=&\sum_{A\ni 0,x} z^{|A|+1} \indic{0\dbct{A} x} \indic{\ve[\iota]\in A} \left( 1 -  \indic{x-\ve[\kappa]\nin A}\right)\nnb
&-\indic{x-\ve[\kappa]=\ve[\iota]}\sum_{A\ni 0,x} z^{|A|+1} \indic{0\dbct{A} x} \indic{x\neq 0}\indic{x-\ve[\kappa]\nin A},
\end{align}

and conclude the desired relation for $x\neq \ve[\iota]+\ve[\kappa]$, with $\iota\neq-\kappa$. For the remaining case $x=\ve[\iota]+\ve[\kappa]$ we compute

\begin{align}
zg_z \Xi^{\ssc[0],\iota}(\ve[\iota]+\ve[\kappa])-\Pi^{\ssc[0],\iota,\kappa}(\ve[\iota]+\ve[\kappa])
=&\sum_{A\ni 0,\ve[\iota]+\ve[\kappa]} z^{|A|+1} \indic{0\dbct{A} \ve[\iota]+\ve[\kappa]} \left(\indic{\ve[\iota]\in A} - \indic{\ve[\iota]\nin A}\right).
\end{align}

To verify that this is positive for all $d\geq 30$, we use the following very helpful rearrangement

\begin{align}
&\indic{\ve[\iota]\in A} - \indic{\ve[\iota]\nin A}=1 - \indic{\ve[\kappa]\nin A}- \indic{\ve[\iota]\nin A},\\
&=1 -\indic{\ve[\iota]\nin A\ni \ve[\kappa]}-\indic{\ve[\kappa]\nin A\ni \ve[\iota]}-2\indic{\ve[\iota],\ve[\kappa]\in A}
=\indic{\ve[\iota],\ve[\kappa]\in A}-\indic{\ve[\iota],\ve[\kappa]\nin A},\nn
\end{align}

so that

\begin{align}
\lbeq{tech-cond-prob-iota-kappa}
zg_z \Xi^{\ssc[0],\iota}(\ve[\iota]+\ve[\kappa])-\Pi^{\ssc[0],\iota,\kappa}(\ve[\iota]+\ve[\kappa])
=&\sum_{A\ni 0,\ve[\iota]+\ve[\kappa]} z^{|A|+1} \indic{0\dbct{A} \ve[\iota]+\ve[\kappa]} \left(\indic{\ve[\iota],\ve[\kappa]\in A} - \indic{\ve[\iota], \ve[\kappa]\nin A}\right).
\end{align}

While it is to be expected that this quantity is indeed positive, we could not find a direct injection between the different classes of LAs with the same number of bonds. Thus, we have to resort to a numerical verification of the inequality, using numerical upper and lower bounds.
We add this extra numerical verification in the LA mathematica notebook (see <cit.>), alongside three other numerical conditions that need to be verified for the analysis of [12], alike $\hat G_z(k)\geq 0$ for all $k$.

Alternatively, this condition could be dropped, at the expense of having to prove separate bounds for $\Pi^{\ssc[0],\iota,\kappa}(x)$, rather than relying on this simple relation to bound $\Pi^{\ssc[0],\iota,\kappa}(x)$ in terms of $ \Xi^{\ssc[0],\iota}(x)$. We refrain from following this route.

<cit.> Growth at the critical point. The functions $z\mapsto \hat \Xi_z(k),z\mapsto \hat  \Xi^{\iota}_z(k),z\mapsto \hat \Psi^{\kappa}_z(k),z\mapsto \hat  \Pi^{\iota,\kappa}_z(k)$ are continuous for $z\in(0,z_c)$. Further, let $\Gamma_1,\Gamma_2,\Gamma_3\geq 0$ be such that $f_i(z)\leq \Gamma_i$ and assume that <cit.> holds. Then, the functions stated above are left-continuous in $z_c$ with a finite limit when $z\nearrow z_c$ for all $x\in\Zd$. Further, for technical reasons, we assume that $z_c<1/2$.

The two-point function $G_z$ is defined as a power series that is clearly continuous in $z$ within its radius of convergence $z_c$. Since $G^\iota_z\leq G_z$, it has a radius of convergence that is at least $z_c$. The coefficients $\Xi^{\ssc[N]}_z,\Xi^{{\ssc[N]},\iota}_z,\Psi^{{\ssc[N]},\iota}_z$ and $\Pi^{{\ssc[N]},\iota,\kappa}_z$ are also power series and can be bounded in terms of many two-point functions. Since $\sum_x G_z(x)<\infty$ for all $z<z_c$, also the radii of convergence of these coefficients are at least $z_c$. [We believe that $G^\iota_z$ and the NoBLE coefficients have the same radii of convergence as $G_z$, but that is irrelevant here.]
Assuming that the bootstrap functions are uniformly bounded, all stated functions are uniformly bounded as well, which implies that they are left-continuous at $z_c$ by Abel's theorem.

To prove that $z_c<1/2$ consider a simple random walk that only takes steps in the positive direction. After $n$ steps there are $d^n$ possible trajectories, each being also a LT, so that $t_n(0)\geq d^n$.
As $z_c$ is defined as the radius of convergence of $\chi$ (see defLTLAsusceptibility), and

\begin{align*}
\chi(1/d)&=\hat {\bar G}_{1/d}(0)=\sum_{n=0}^\infty \sum_{x} t_n(x)d^{-n}\geq \sum_{n=0}^\infty t_n(0)d^{-n}= \sum_{n=0}^\infty d^n d^{-n}=\infty,
\end{align*}

we know that $z_c\leq 1/d<1/2$.

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[line width=1 pt] (1,0) to (1,1);
[line width=1 pt] (0,1) to (0.5,1.25);
[line width=1 pt] (0.5,1.25) to (1,1);

[line width=1 pt] (0.2,0) to (-0.2,0);
[line width=1 pt] (0,0.2) to (0,-0.2);

[line width=2 pt] (0,0) to (1,0);
[line width=2 pt] (1,0) to (1,1);
[line width=1 pt] (0,1) to (0.5,1.25);
[line width=1 pt] (0.5,1.25) to (1,1);

[line width=1 pt] (0,0-0.2) to (0,1+0.2);
[line width=1 pt] (-.3,0-0.2) to (0,0-0.2);
[line width=1 pt] (-.3,1+0.2) to (0,1+0.2);
[above]   at(0,1.4)      $N-2$;

[line width=2 pt] (0,0) to (1,0);
[line width=1 pt] (1,0) to (1,1);
[line width=1 pt] (0,1) to (1,1);

[line width=1pt,auto,scale=#1]
[-](0,0) to  (0,2);
[line width=2pt,-](0,0) to  (2,0);
[-](0,2) to    (2,2);
[line width=2pt](2,0) to    (2,2);
(0,2) circle (3pt);
(2,0) circle (3pt);
(2,2) circle (3pt);
(0,0) circle (3pt);
[left]   at(0,0)      $0$;
[left]   at(0,2)      $v$;
[right]  at (2,0)     $y$;
[right]  at (2,2)     $x$;
[right]  at (2,1)     $s$;

[line width=1pt,auto,scale=#1]
[-](0,0) to  (0,2);
[line width=2pt,-](0,0) to  (2,0);
[-](0,2) to    (2,2);
[-](2,0) to    (2,2);
(0,2) circle (3pt);
(2,0) circle (3pt);
(2,2) circle (3pt);
(0,0) circle (3pt);
[left]   at(0,0)      $0$;
[left]   at(0,2)      $v$;
[right]  at (2,0)     $x$;
[right]  at (2,2)     $y$;
[right]  at (2,1)     $d_R$;

[line width=1pt,auto,scale=#1]
[-](0,0) to  (0,2);
[line width=2pt,-](0,0) to  (2,0);
[-](0,2) to    (2,2);
[line width=2pt](2,0) to    (2,2);
(0,2) circle (3pt);
(2,0) circle (3pt);
(2,2) circle (3pt);
(0,0) circle (3pt);
(0.7,0) circle (4pt);

[left]   at(0,0)      $0$;
[below]   at(0.7,0)   $\ve[\iota]$;
[left]   at(0,2)      $v$;
[right]  at (2,0)     $y$;
[right]  at (2,2)     $x$;
[right]  at (2,1)     $s$;

[line width=1pt,auto,scale=#1]
[-](0,0) to  (0,2);
[line width=2pt,-](0,0) to  (2,0);
[-](0,2) to    (2,2);
[-](2,0) to    (2,2);
(0,2) circle (3pt);
(0.7,0) circle (4pt);
(2,0) circle (3pt);
(2,2) circle (3pt);
(0,0) circle (3pt);
[left]   at(0,0)      $0$;
[below]   at(0.7,0)   $\ve[\iota]$;
[left]   at(0,2)      $v$;
[right]  at (2,0)     $x$;
[right]  at (2,2)     $y$;
[right]  at (2,1)     $d_R$;

[line width=1pt,auto,scale=#1]
[-](0,0) to  (0.25,1.2);
[-](0.25,1.2) to  (1.4,2);
[-](1.4,2) to  (2.5,2);
[line width=2pt,-](0,0) to    (2.5,0);
[line width=2pt,-](2.5,0) to    (2.5,2);
[-](0.25,1.2) to  [out=135,in=90]   (-0.8,0);
(0,0) circle (3pt);
(0.25,1.2) circle (3pt);
(1.4,2) circle (3pt);
(2.5,2) circle (3pt);
(2.5,0) circle (3pt);
(-0.8,0) circle (3pt);
[below]   at(-0.8,0)      $\ve[\iota]$;
[below]   at(0,0)      $0$;
[above]   at(0.25,1.2)      $u$;
[left]   at(1.5,1.5)      $v$;
[right]  at (2.5,0)     $y$;
[right]  at (2.5,2)     $x$;
[right]  at (2.5,1)     $s$;

[line width=1pt,auto,scale=#1]
[-](0,0) to  (0.25,1.2);
[-](0.25,1.2) to  (1.4,2);
[-](1.4,2) to  (2.5,2);
[line width=2pt,-](0,0) to    (2.5,0);
[-](2.5,0) to    (2.5,2);
[-](0.25,1.2) to  [out=135,in=90]   (-0.8,0);
(0,0) circle (3pt);
(0.25,1.2) circle (3pt);
(1.4,2) circle (3pt);
(2.5,2) circle (3pt);
(2.5,0) circle (3pt);
(-0.8,0) circle (3pt);
[below]   at(-0.8,0)      $\ve[\iota]$;
[below]   at(0,0)      $0$;
[above]   at(0.25,1.2)      $u$;
[left]   at(1.5,1.5)      $v$;
[right]  at (2.5,0)     $x$;
[right]  at (2.5,2)     $y$;
[right]  at (2.5,1)     $d_R$;

[line width=1pt,auto,scale=#1]
(0,0)– (2,0)– (1,1.5)– cycle;
(0,0) circle (3pt);
(2,0) circle (3pt);
(1,1.5) circle (3pt);
[left]   at(0,0)       $0$;
[right]   at (2,0)       $x$;
[left]   at (1,1.5)       $v$;

[line width=1pt,auto,scale=#1]
[-](0,0) to [out=90,in=90]    (2,0);
[-](0,0) to [out=270,in=270]    (2,0);
(0,0) circle (3pt);
[left]   at(0,0)      $0$;
[right]  at (2,0)   $\geq 4$;

[line width=1pt,auto,scale=#1]
[-](0,1.5)– (2,0);
[line width=1pt,-] (2,0) –(0,0);
[line width=1pt,-] (0,0) –(0,1.5);
(0,0) circle (3pt);
(0,1.5) circle (3pt);
(2,0) circle (3pt);
[left]   at(0,0)      $0$;
[right]  at (2,0)     $x$;
[left]   at (0,1.5)   $v$;
[left]   at (0,.725)  $=1$;
[below]  at (1,0)     $=1$;
[above]  at (1.5,0.7)     $\geq 2$;

[-](4,0)– (6,0);
[-](5,1.5)– (4,0);
[-](5,1.5)– (6,0);
(4,0) circle (3pt);
(5,1.5) circle (3pt);
(6,0) circle (3pt);
[left]   at(4,0)       $0$;
[right]   at (6,0)       $x$;
[left]   at (5,1.5)      $v$;
[below]  at (5,0)        $=1$;
[left]   at (4.6,.9)     $\geq 2$;
[right]  at (5.4,0.9)    $\geq 1$;

[line width=1pt,auto,scale=#1]
[-](0,0) to [out=90,in=90]    (3,0);
[-](0,0) to [out=270,in=270]    (3,0);
(0,0) circle (3pt);
(3,0) circle (3pt);
[left]   at(0,0)      $0$;
[right]  at (3,0)     $x$;
[above]  at (1,0.7)     $\geq 3$;
[below]  at (2,-0.7)     $=1$;
[-](5,0) to [out=90,in=90]    (8,0);
[-](5,0) to [out=270,in=270]    (8,0);
(5,0) circle (3pt);
(8,0) circle (3pt);
[left]   at(5,0)      $0$;
[right]  at (8,0)     $x$;
[above]  at (6,0.7)     $\geq 2$;
[below]  at (7,-0.7)     $\geq 2$;

[line width=1pt,auto,scale=#1]
[-] (0,0)–(1,1.5);
[-] (1,1.5)– (2,0);
[-] (2,0)–(0,0);
(0,0) circle (3pt);
(1,1.5) circle (3pt);
(2,0) circle (3pt);
[left]   at(0,0)       $0$;
[right]   at (2,0)       $x$;
[left]   at (1,1.5)      $v$;
[below]  at (1,0)        $\geq 2$;
[left]   at (0.6,.9)     $\geq 1$;
[right]  at (1.4,0.9)    $\geq 1$;

[line width=1pt,auto,scale=#1]
[-](0,0) to [out=90,in=90]    (3,0);
[-](0,0) to [out=270,in=270]    (3,0);
(0,0) circle (3pt);
(3,0) circle (3pt);
[left]   at(0,0)      $0$;
[right]  at (3,0)     $y$;
[above]  at (1,0.7)     $\geq 3$;
[below]  at (2,-0.7)     $= 1$;

[line width=1pt,auto,scale=#1]
[-](0,0) to [out=90,in=180]    (2,2);
[-](2,0) to    (2,2);
[-](0,0) to    (2,0);
(0,0) circle (3pt);
(2,0) circle (3pt);
(2,2) circle (3pt);
[left]   at(0,0)      $0$;
[right]  at (2,0)     $y$;
[right]  at (2,2)     $x$;
[below]  at (0.6,0.3)  $\geq 2$;
[above]  at (1,1.8)   $= 1$;
[right]  at (2,1)     $=1$;

[-](4,0) to [out=90,in=180]    (6,2);
[-](6,0) to    (6,2);
[-](4,0) to    (6,0);
(4,0) circle (3pt);
(6,0) circle (3pt);
(6,2) circle (3pt);
[left]   at(4,0)      $0$;
[right]  at (6,0)     $y$;
[right]  at (6,2)     $x$;
[above]  at (4.4,1.5)  $\geq 2$;
[below]  at (5,-0.2)  $\geq 1$;
[right]  at (6,1)     $=1$;

[line width=1pt,auto,scale=#1]
[-](0,0) to  (0,2);
[-](0,0) to  (2,0);
[-](0,2) to    (2,2);
[-](2,0) to    (2,2);
(0,2) circle (3pt);
(2,0) circle (3pt);
(2,2) circle (3pt);
(0,0) circle (3pt);
[left]   at(0,0)      $0$;
[left]   at(0,2)      $v$;
[right]  at (2,0)     $y$;
[right]  at (2,2)     $x$;
[left]  at (0,1)      $\geq 1$;
[below]  at (1,0)     $\geq 1$;
[above]  at (1,1.8)   $\geq 1$;
[right]  at (2,1)     $=1$;

[line width=1pt,auto,scale=#1]
[-](0,1.5)– (2,0);
[line width=1pt,-] (2,0) –(0,0);
[line width=1pt,-] (0,0) –(0,1.5);
(0,0) circle (3pt);
(0,1.5) circle (3pt);
(2,0) circle (3pt);
[left]   at(0,0)      $0$;
[right]  at (2,0)     $x$;
[left]   at (0,1.5)   $v$;
[left]   at (0,.725)  $=1$;
[below]  at (1,0)     $=1$;
[above]  at (1.5,0.7)     $\geq 2$;

[-](5,1.5)– (6,0);
[line width=1pt,-] (4,0) –(6,0);
(4,0) circle (3pt);
(5,1.5) circle (3pt);
(6,0) circle (3pt);
[left]   at(4,0)       $0$;
[right]   at (6,0)       $x$;
[left]   at (5,1.5)      $v$;
[below]  at (5,0)        $=1$;
[left]   at (4.6,.9)     $\geq 2$;
[right]  at (5.4,0.9)    $\geq 1$;

[line width=1pt,auto,scale=#1]
[-](0,0) to [out=90,in=90]    (3,0);
[-](0,0) to [out=270,in=270]  (3,0);
(0,0) circle (3pt);
(3,0) circle (3pt);
[left]   at(0,0)      $0$;
[right]  at (3,0)     $y$;
[above]  at (1,0.7)     $\geq 1$;
[below]  at (2,-0.7)     $\geq 2$;

[line width=1pt,auto,scale=#1]
[-](4,0) to [out=90,in=180]    (6,2);
[-](6,0) to    (6,2);
[-](4,0) to    (6,0);
(4,0) circle (3pt);
(6,0) circle (3pt);
(6,2) circle (3pt);
[left]   at(4,0)      $0$;
[right]  at (6,0)     $y$;
[right]  at (6,2)     $x$;
[above]  at (3.5,0.5) $\geq 1$;
[below]  at (5,-0.3)  $\geq 1$;
[right]  at (6,1)     $\geq 2$;

[line width=1pt,auto,scale=#1]
[-](4,0) to     (4,2);
[-](4,2) to     (6,2);
[-](6,0) to    (6,2);
[-](4,0) to    (6,0);
(4,0) circle (3pt);
(4,2) circle (3pt);
(6,0) circle (3pt);
(6,2) circle (3pt);
[left]   at(4,0)      $0$;
[left]   at(4,2)      $v$;
[right]  at (6,0)     $y$;
[right]  at (6,2)     $x$;
[left]  at (4,1) $\geq 1$;
[above]  at (5,2) $\geq 0$;
[below]  at (5,-0.2)  $\geq 1$;
[right]  at (6,1)     $\geq 2$;

[line width=1pt,auto,scale=#1]
[-](4,0) to [out=90,in=180]    (6,2);
[-](6,0) to    (6,2);
[-](4,0) to    (6,0);
(4,0) circle (3pt);
(6,0) circle (3pt);
(6,2) circle (3pt);
[left]   at(4,0)      $0$;
[right]  at (6,0)     $x$;
[right]  at (6,2)     $y$;
[above]  at (4.4,1.5) $\geq 2$;
[below]  at (5,-0.2)  $= 1$;
[right]  at (6,1)     $=1$;

[line width=1pt,auto,scale=#1]
[-](4,0) to     (4,2);
[-](4,2) to     (6,2);
[-](6,0) to    (6,2);
[-](4,0) to    (6,0);
(4,0) circle (3pt);
(4,2) circle (3pt);
(6,0) circle (3pt);
(6,2) circle (3pt);
[left]   at(4,0)      $0$;
[left]   at(4,2)      $v$;
[right]  at (6,0)     $x$;
[right]  at (6,2)     $y$;
[left]  at (4,1) $\geq1$;
[above]  at (5,2) $\geq1$;
[below]  at (5,-0.2)  $=1$;
[right]  at (6,1)     $=1$;

[line width=1pt,auto,scale=#1]
[-](4,0) to [out=90,in=180]    (6,2);
[-](6,0) to    (6,2);
[-](4,0) to    (6,0);
(4,0) circle (3pt);
(6,0) circle (3pt);
(6,2) circle (3pt);
[left]   at(4,0)      $0$;
[right]  at (6,0)     $x$;
[right]  at (6,2)     $y$;
[above]  at (4.4,1.5) $\geq 1$;
[below]  at (5,-0.2)  $\geq 2$;
[right]  at (6,1)     $=1$;

[line width=1pt,auto,scale=#1]
[-](4,0) to     (4,2);
[-](4,2) to     (6,2);
[-](6,0) to    (6,2);
[-](4,0) to    (6,0);
(4,0) circle (3pt);
(4,2) circle (3pt);
(6,0) circle (3pt);
(6,2) circle (3pt);
[left]   at(4,0)      $0$;
[left]   at(4,2)      $v$;
[right]  at (6,0)     $x$;
[right]  at (6,2)     $y$;
[left]  at (4,1) $\geq 1$;
[above]  at (5,2) $\geq 0$;
[below]  at (5,-0.2)  $\geq 2$;
[right]  at (6,1)     $=1$;

[line width=1pt,auto,scale=#1]
[-](4,0) to [out=90,in=180]    (6,2);
[-](6,0) to    (6,2);
[-](4,0) to    (6,0);
(4,0) circle (3pt);
(6,0) circle (3pt);
(6,2) circle (3pt);
[left]   at(4,0)      $0$;
[right]  at (6,0)     $x$;
[right]  at (6,2)     $y$;
[above]  at (4.4,1.5) $\geq 1$;
[below]  at (5,-0.2)  $\geq 1$;
[right]  at (6,1)     $\geq 2$;

[line width=1pt,auto,scale=#1]
[-](4,0) to     (4,2);
[-](4,2) to     (6,2);
[-](6,0) to    (6,2);
[-](4,0) to    (6,0);
(4,0) circle (3pt);
(4,2) circle (3pt);
(6,0) circle (3pt);
(6,2) circle (3pt);
[left]   at(4,0)      $0$;
[left]   at(4,2)      $v$;
[right]  at (6,0)     $x$;
[right]  at (6,2)     $y$;
[left]  at (4,1) $\geq1$;
[above]  at (5,2) $\geq0$;
[below]  at (5,-0.2)  $\geq 1$;
[right]  at (6,1)     $\geq 2$;

[line width=1pt,auto,scale=#1]
[-](0,0) to [out=90,in=90]    (3,0);
[-](0,0) to     (3,0);
(0,0) circle (3pt);
(3,0) circle (3pt);
[left]   at(0,0)      $0,x$;
[right]  at (3,0)     $\ve[\iota]$;
[above]  at (1,0.7)     $\geq 3$;
[below]  at (2,0)     $=1$;

[line width=1pt,auto,scale=#1]
[-](0,0) to (2,0);
[-](0,0) to (0,1.5);
[-](0,1.5) to  [out=0,in=90]   (2,0);
(0,0) circle (3pt);
(2,0) circle (3pt);
(0,1.5) circle (3pt);
[left]   at(0,0)      $0$;
[right]  at (2,0)     $\ve[\iota]$;
[left]   at (0,1.5)      $v$;
[left]  at (0,1)      $=1$;
[below]  at (1,0)     $=1$;
[right]  at (1.2,1.5)     $\geq2$;

[-] (4,0)to(5,1.5);
[-] (5,1.5) to (6,0);
[-] (4,0) –(6,0);
(4,0) circle (3pt);
(5,1.5) circle (3pt);
(6,0) circle (3pt);
[left]   at(4,0)       $0$;
[right]   at (6,0)       $\ve[\iota]$;
[left]   at (5,1.5)      $v$;
[below]  at (5,0)        $=1$;
[left]   at (4.6,.9)     $\geq 2$;
[right]  at (5.4,0.9)    $\geq 1$;

[line width=1pt,auto,scale=#1]
[-](0,0) to (2,0);
[-](0,0) to (0,1.5);
[-](0,1.5) to  [out=0,in=90]   (2,0);
(0,0) circle (3pt);
(2,0) circle (3pt);
(0,1.5) circle (3pt);
[left]   at(0,0)      $0$;
[right]  at (2,0)     $\ve[\iota]$;
[left]   at(0,1.5)      $x$;
[left]  at (0,0.6)      $=1$;
[below]  at (1,0)     $=1$;
[right]  at (1.2,1.5)     $\geq2$;

[-] (4,0)to(5,1.5);
[-] (5,1.5) to (6,0);
[-] (4,0) –(6,0);
(4,0) circle (3pt);
(5,1.5) circle (3pt);
(6,0) circle (3pt);
[left]   at(4,0)       $0$;
[right]   at (6,0)       $\ve[\iota]$;
[left]   at (5,1.5)      $x$;
[below]  at (5,0)        $=1$;
[left]   at (4.6,.9)     $\geq 2$;
[right]  at (5.4,0.9)    $\geq 1$;

[line width=1pt,auto,scale=#1]
[-](4,0) to     (4,2);
[-](4,2) to     (6,2);
[-](6,0) to    (6,2);
[-](4,0) to    (6,0);
(4,0) circle (3pt);
(4,2) circle (3pt);
(6,0) circle (3pt);
(6,2) circle (3pt);
[left]   at(4,0)      $0$;
[left]   at(4,2)      $v$;
[right]  at (6,0)     $\ve[\iota]$;
[right]  at (6,2)     $x$;
[left]  at (4,1) $\geq1$;
[above]  at (5,2) $\geq 1$;
[below]  at (5,-0.2)  $=1$;
[right]  at (6,1)     $\geq 1$;

[line width=1pt,auto,scale=#1]
[-](0,0) to (2,0);
[-](0,0) to (0,1.5);
[-](0,1.5) to  [out=0,in=90]   (2,0);
(0,0) circle (3pt);
(2,0) circle (3pt);
(0,1.5) circle (3pt);
[left]   at(0,0)      $0$;
[below]  at (2,0)     $\ve[\iota]$;
[left]   at(0,1.5)      $v$;
[left]  at (0,0.6)      $=1$;
[below]  at (0.6,0)     $=1$;
[right]  at (1.2,1.5)     $\geq2$;

[-] (5,0) to(6,1.5);
[-](6,1.5) to (7,0);
[-] (5,0) –(7,0);
(5,0) circle (3pt);
(6,1.5) circle (3pt);
(7,0) circle (3pt);
[left]   at(5,0)       $0=y$;
[right]   at (7,0)       $\ve[\iota]$;
[left]   at (6,1.5)      $v$;
[below]  at (6,0)        $=1$;
[left]   at (5.6,.9)     $\geq 2$;
[right]  at (6.4,0.9)    $\geq 1$;

[line width=1pt,auto,scale=#1]
[-](-1,0) to (1,0);
[-](-1,0) to [out=270,in=180] (0,-1);
[-](1,0) to [out=270,in=0] (0,-1);
(-1,0) circle (3pt);
(1,0) circle (3pt);
[left]   at(-1,0)      $0=x$;
[above]  at (1.2,-0.1)     $\ve[\iota]$;
[above]  at (-.3,0)    $=1$;
[right]  at (0.6,-0.8)     $\geq 3$;

[line width=1pt,auto,scale=#1]
[-](-1,0) to (1,0);
[-](-1,0) to [out=90,in=180] (1,1.5);
[-](1,1.5) to     (1,0);
(-1,0) circle (3pt);
(1,1.5) circle (3pt);
(1,0) circle (3pt);
[left]   at(-1,0)      $0$;
[below]  at (1.3,0)     $\ve[\iota]$;
[right]   at(1,1.5)   $x$;
[right]  at (1,0.7)    $=1$;
[below]  at (0,0)     $=1$;
[left]  at (0,1.4)     $\geq2$;

[line width=1pt,auto,scale=#1]
[-] (7,0)–(5,0);
[-] (5,0)–(5,1.5);
[-] (5,1.5)  – (7,1.5);
[-] (7,0) –(7,1.5);
(5,0) circle (3pt);
(5,1.5) circle (3pt);
(7,1.5) circle (3pt);
(7,0) circle (3pt);
[left]   at(5,0)       $0$;
[right]   at (7,0)       $\ve[\iota]$;
[right]   at (7,1.5)      $x$;
[left]   at (5,1.5)      $v$;
[below]  at (6,0)        $=1$;
[left]   at (5,.9)     $\geq 1$;
[above]   at (6,1.5)     $\geq 1$;
[right]  at (7,0.9)    $= 1$;

[line width=1pt,auto,scale=#1]
[-](0,0) to (2,0);
[-](0,0) to (0,1.5);
[-](0,1.5) to  [out=0,in=90]   (2,0);
(0,0) circle (3pt);
(2,0) circle (3pt);
(0,1.5) circle (3pt);
[left]   at(0,0)      $0=x$;
[below]  at (2,0)     $\ve[\iota]$;
[left]   at(0,1.5)     $y$;
[left]  at (0,0.6)      $=1$;
[below]  at (0.6,0)     $=1$;
[right]  at (1.2,1.5)   $\geq2$;

[line width=1pt,auto,scale=#1]
[-](0,0) to (2,0);
[-](0,0) to (0,2);
[-](2,2) to (0,2);
[-](2,2) to  (2,0);
(0,0) circle (3pt);
(2,0) circle (3pt);
(2,2) circle (3pt);
(0,2) circle (3pt);
[left]   at(0,0)      $0$;
[right]  at (2,0)     $\ve[\iota]$;
[left]   at(0,2)     $x$;
[right]   at(2,2)     $y$;
[left]  at (0,1)      $\geq 1$;
[above]  at (1,2)      $=1$;
[below]  at (1,0)     $=1$;
[right]  at (2,1)   $\geq1$;

[line width=1pt,auto,scale=#1]
[-](0,0) to (-1+0.25,1.5);
[-](0,0) to (2,0);
[-](2,0) to (3-0.25,1.5);
[-](3-0.25,1.5) to  (1,2.5);
[-](-1+0.25,1.5) to (1,2.5);
(0,0) circle (3pt);
(2,0) circle (3pt);
(3-0.25,1.5) circle (3pt);
(-1+0.25,1.5) circle (3pt);
(1,2.5) circle (3pt);

[left]   at(0,0)      $0$;
[right]  at (2,0)     $\ve[\iota]$;
[above]   at(1,2.5)     $x$;
[right]   at(2.75,1.5)     $y$;
[left]   at(-0.75,1.5)     $v$;

[left]  at (-.5,0.6)      $\geq 0$;
[above]  at (2.3,2.1)      $=1$;
[below]  at (1,0)     $=1$;
[right]  at (2.5,0.6)   $\geq1$;
[above]  at (0,2.1)   $\geq 1$;

[line width=1pt,auto,scale=#1]
[-](4+0.8,0-0.8) to [out=90,in=180]    (6+0.8,2-0.8);
[-](6+0.8,0-0.8) to    (6+0.8,2-0.8);
[-](4+0.8,0-0.8) to    (6+0.8,0-0.8);
(4+0.8,0-0.8) circle (3pt);
(6+0.8,0-0.8) circle (3pt);
(6+0.8,2-0.8) circle (3pt);
[left]   at(4+0.8,0-0.8)      $0$;
[right]  at (6+0.8,0-0.8)     $\ve[\iota]$;
[right]  at (6+0.8,2-0.8)     $y$;
[above]  at (3.8+0.8,1-0.8) $\geq 2$;
[below]  at (5+0.8,-0.1-0.8)  $= 1$;
[right]  at (6+0.8,1-0.8)     $\geq0$;

[line width=1pt,auto,scale=#1]
[-](4,0) to (4,2);
[-](4,2) to (6,2);
[-](6,0) to    (6,2);
[-](4,0) to    (6,0);
(4,0) circle (3pt);
(6,0) circle (3pt);
(6,2) circle (3pt);
(6,0) circle (3pt);
[left]   at (4,0)      $0$;
[left]   at (4,2)      $x$;
[right]  at (6,0)     $\ve[\iota]$;
[right]  at (6,2)     $y$;
[above]  at (5,2)  $\geq 2$;
[left]  at (4,1)  $\geq 1$;
[below]  at (5,-0.1)  $= 1$;
[right]  at (6,1)     $\geq 0$;

[line width=1pt,auto,scale=#1]
[-](0,0) to (-1+0.25,1.5);
[-](0,0) to (2,0);
[-](2,0) to (3-0.25,1.5);
[-](3-0.25,1.5) to  (1,2.5);
[-](-1+0.25,1.5) to (1,2.5);
(0,0) circle (3pt);
(2,0) circle (3pt);
(3-0.25,1.5) circle (3pt);
(-1+0.25,1.5) circle (3pt);
(1,2.5) circle (3pt);

[left]   at(0,0)      $0$;
[right]  at (2,0)     $\ve[\iota]$;
[above]   at(1,2.5)     $x$;
[right]   at(2.75,1.5)     $y$;
[left]   at(-0.75,1.5)     $v$;

[left]  at (-.5,0.6)      $\geq 0$;
[above]  at (2.3,2.1)      $\geq 2$;
[below]  at (1,0)     $=1$;
[right]  at (2.5,0.6)   $\geq0$;
[above]  at (0,2.1)   $\geq 1$;

[line width=1pt,auto,scale=#1]
[-](-1,0) to (1,0);
[-](1,0) to [out=90,in=0] (-1,1.5);
[-](-1,1.5) to     (-1,0);
(-1,0) circle (3pt);
(-1,1.5) circle (3pt);
(1,0) circle (3pt);
[left]   at(-1,0)      $0=y$;
[below]  at (1.3,0)     $\ve[\iota]$;
[left]   at(-1,1.5)   $x$;
[left]  at (-1,0.7)    $=1$;
[below]  at (0,0)     $=1$;