# The radius of a self-repelling star polymer

Carl Mueller  Carl Mueller
Department of Mathematics
University of Rochester
Rochester, NY 14627<EMAIL_ADDRESS>and  Eyal Neuman  Eyal
Neuman
Department of Mathematics
Imperial College
London, UK<EMAIL_ADDRESS>
###### Abstract.

We study the effective radius of weakly self-avoiding star polymers in one,
two, and three dimensions. Our model includes $N$ Brownian motions up to time
$T$, started at the origin and subject to exponential penalization based on
the amount of time they spend close to each other, or close to themselves. The
effective radius measures the typical distance from the origin. Our main
result gives estimates for the effective radius where in two and three
dimensions we impose the restriction that $T\leq N$. One of the highlights of
our results is that in two dimensions, we find that the radius is proportional
to $T^{3/4}$, up to logarithmic corrections.

###### Key words and phrases:

Polymers, self-avoiding, star polymers

###### 2020 Mathematics Subject Classification:

Primary, 60G60; Secondary, 60G15.

CM was partially supported by Simons grant 513424.

##  1\. Introduction

Random polymer models have caught the imagination of many mathematicians.
Polymers are all around us, and their behavior presents attractive
mathematical challenges, many of which still defy solution. See Doi and
Edwards [8] for a wide-ranging treatment from the physical point of view, and
Madras and Slade [15], den Hollander [7], Giacomin [11], and Bauerschmidt et.
al. [3] for rigorous mathematical results. Van der Hofstad and König [20] and
van der Hofstad, den Hollander, and König [18], [19] discuss the one-
dimensional situation for the related Edwards model. Bauerschmidt, Brydges,
and Slade [2] have recently developed a rigorous version of the
renormalization group and applied it to the four-dimensional situation.

In continuous time, we can view a polymer as a Brownian motion with
penalization for self-intersections. Here the time parameter represents
distance along the polymer. For $T>0$, let $(B_{t})_{t\in[0,T]}$ be a standard
Brownian motion in $\mathbb{R}^{d}$, defined on a filtered probability space
$(\Omega,(\mathcal{F}_{t})_{t\in[0,T]},\mathcal{F},P_{T})$. For a probability
measure $P$, we write $E^{P}$ for the corresponding expectation. Since
Brownian motion does not have self-intersections in high dimensions, we study
close approaches instead. For any $r>0$ let
$\mathbf{B}_{r}(x)\subset\mathbb{R}^{d}$ be the open ball of radius $r$
centered at $x\in\mathbb{R}^{d}$, and define

$L_{T}(x)=L_{d,T}(x):=\int_{0}^{T}\mathbf{1}_{\mathbf{B}_{1}(x)}(B_{t})dt$

for $T>0$. For $\beta>0$, the typical penalization term is

$\mathcal{E}_{T}=\mathcal{E}_{d,\beta,T}:=\exp\left(-\beta\int_{\mathbb{R}^{d}}L_{T}(x)^{2}dx\right).$

Then we define the penalized measure as

$\begin{split}Q_{T}(A)&=Q_{d,\beta,T}(A):=\frac{1}{Z_{T}}E^{P_{T}}\left[\mathbf{1}_{A}\mathcal{E}_{T}\right]\\\
Z_{T}&=Z_{d,\beta,T}=E^{P_{T}}\left[\mathcal{E}_{T}\right]\end{split}$

for $A\in\mathcal{F}$. For simplicity of notation, we have suppressed some of
the subscripts.

With these definitions, we call our process weakly self-avoiding Brownian
motion.

Note that all of the above quantities depend implicitly on $d$ and all but
$P,\mathbf{B},(B_{t}),L$ depend on $\beta$ as well. For simplicity of notation
we suppress these dependencies, and we will use similar simplified notation
throughout the paper. Furthermore, $C$ will stand for a constant which could
change from line to line, and may also depend on $d$.

One of the most important problems about weakly self-avoiding Brownian motion
is to study the radius of the polymer, often defined as the standard deviation
of the end-to-end distance,

$R_{T}=R_{d,\beta,T}=\left(E^{Q_{T}}\left[B_{T}^{2}\right]\right)^{1/2}.$

A well-known conjecture from physics states that there exists a scaling
exponent $\nu_{d}$ not depending on $\beta$ such that, in some unspecified
sense,

$R_{d,\beta,T}\approx T^{\nu_{d}}$

as $T\to\infty$. All that is rigorously known is that $\nu_{1}=1$ (Bolthausen
[5], Greven and den Hollander [12]) and that $\nu_{d}=1/2$ for $d\geq 5$ (Hara
and Slade [13]). It is believed that $\nu_{2}=3/4$, and there are connections
to SLE8/3 (see Lawler, Schramm, and Werner [14]). This conjecture has received
enormous attention, and Duminil-Copin and Smirnov [9], page 9, write “The
derivation of these exponents seems to be one of the most challenging problems
in probability.” In [3] Section 1.5.2, we learn that “Almost nothing is known
rigorously about $\nu$ in dimensions 2, 3, 4. It is an open problem to show
that the mean-square displacement grows at least as rapidly as simple random
walk, and grows more slowly than ballistically”. In our language, this means
that for $d\in\\{2,3,4\\}$ it has not been proved that $\nu_{d}\geq 1/2$ or
$\nu_{d}<1$. One of the highlights of the present work is that we do obtain
the exponent $3/4$ in $d=2$, see Theorem 1.1. Of course this does not settle
the above conjecture, and our methods are unlikely to settle it.

In the real world, many polymers are branched and do not consist of a single
strand. van der Hofstad and König [20], (Section 3.1 pages 16 \- 18) give a
short discussion of branched polymers taking values in $\mathbb{R}$, and
present a conjecture for the growth of the radius. As far as we know, the
conjecture is still open. Slade and van der Hofstad [16] use the lace
expansion to study the radius for branched polymers in high dimensions, and
show that they behave as if there were no self-avoidance.

Since self-avoiding polymers present difficult challenges in low dimensions,
we focus on the case of star polymers, which are not too different from random
walks. Star polymers are polymers joined at the point $t=0$, and there is an
extensive physics literature about them. See the seminal paper of Daoud and
Cotton [6], and for more recent work see [1] and [17].

We now give a brief overview of the results from [6] which are relevant to
this paper, and ask the reader to keep in mind that these results come from
mathematically non-rigorous arguments.

First, the authors formulate a notion of radius relevant to star polymers.
Then, for a given value of $T$, they discuss 3 regions:

  1. (1)

The swollen region far from the origin, where pairs of paths rarely overlap.

  2. (2)

The unswollen region closer to the origin, where many paths overlap.

  3. (3)

The core, which is even closer to the origin.

Our results deal with regions (2) and (3). One of the principal results of [6]
is equation (19), which states that for very long chains, or for high
temperatures,

(1.1) $R\approx N^{3/5}v^{1/5}f^{1/5}\ell.$

To aid the comparison with our results, we give a dictionary for the notation
in the above equation.

Quantity | Our Notation | From [6]
---|---|---
Number of branches | $N$ | $f$
Length along the polymer | $T$ | $N$
Self-avoidance parameter | $\beta$ | $v$
Length of each polymer element | not included | $\ell$

Translating to our notation, (1.1) means, for $d=3$, that

(1.2) $R_{T}\approx\beta^{1/5}N^{1/5}T^{3/5}$

for large values of $T$. Again, this is a nonrigorous physical result. Our
main result, Theorem 1.1, gives a range of values for $R_{T}$ in $d=3$, that
includes (1.2). The physical reasoning in Daoud and Cotton’s paper yields the
following conjecture for the two dimensional model,

(1.3) $R\approx\ \beta^{1/4}N^{1/4}T^{3/4}.$

We refer to Bishop et al. [4] who studied the two dimensional star polymer
model. In Theorem 1.1 we verify (1.3) up to a logarithmic constant.

We will now give rigorous definitions, and redefine the notation $P_{T}$,
$Q_{T}$, $L_{T}(\cdot)$, $\mathcal{E}_{T}$, $R_{T}$ used earlier. Assume that
for each $T>0$ we have a filtered probability space
$(\Omega,\mathcal{F},(\mathcal{F}_{t})_{t\in[0,T]},P_{T})$, and on this space,
for $d\geq 1$, we have a collection
$(B^{(k)}_{t})_{t\in[0,T];\;k\in\\{1,\dots,N\\}}$ of independent adapted
$\mathbb{R}^{d}$-valued standard Brownian motions started at the origin.
Without loss of generality, we can assume that $\Omega=(C[0,T])^{N}$ is
canonical path space for the Brownian motions.

We define a weakly self-avoiding model as follows. For
$N\in\mathbb{N}=\\{1,2,\dots\\}$, $T>0$, $x\in\mathbb{R}^{d}$, and
$\mathbf{B}_{1}(x)\subset\mathbb{R}^{d}$ as before, consider the occupation
measure

(1.4)
$L_{T}(x)=L_{T,d,N}(x):=\sum_{k=1}^{N}|\\{t\in[0,T]:B^{(k)}_{t}\in\mathbf{B}_{1}(x)\\}|$

where $|S|$ denotes the Lebesgue measure of a Borel set $S\subset\mathbb{R}$.
Our penalization factor is defined as

$\mathcal{E}_{T}:=\mathcal{E}_{d,\beta,N,T}=\exp\left(-\beta\int_{\mathbb{R}^{d}}L_{T}(x)^{2}dx\right).$

We define a probability $Q_{T}=Q_{T,d,N,\beta}$ and a normalizing factor
$Z_{T}=Z_{T,d,N,\beta}$ as

(1.5) $\displaystyle
Q_{T}(S):=\frac{1}{Z_{T}}E^{P_{T}}\big{[}\mathbf{1}_{S}\mathcal{E}_{T}\big{]},$
$\displaystyle Z_{T}:=E^{P_{T}}\big{[}\mathcal{E}_{T}\big{]}.$

for $S\in\mathcal{F}_{T}$.

For any set of real numbers $A=\\{a_{1},...,a_{N}\\}$ we denote by
$\textrm{med}(A)$ the median of the set. We define the radius of the star
polymer as follows,

(1.6)
$R_{T}=R_{d,\beta,N,T}:=\textrm{med}\left(\left\\{\sup_{t\in[0,T]}|B^{(k)}_{t}|\,:k=1,...,N\right\\}\right).$

Our goal is to study the behavior of $R_{T}$ under the measure $Q_{T}$.

For convenience we have chosen a different definition of radius than before.
But there are already several definitions of the radius, for example see
Fixman [10].

In order to to study our radius $R_{T}$, we introduce the following events

(1.7) $\begin{split}A^{(<)}_{T,r}&=\\{R_{T}\leq r\\},\\\
A^{(>)}_{T,r}&=\\{R_{T}\geq r\\}.\end{split}$

We will show that for appropriate functions $r_{1}(T),r_{2}(T)$,

$\begin{split}\lim_{T\to\infty}Q_{T}\big{(}A^{(<)}_{T,r_{1}(T)}\big{)}&=0,\\\
\lim_{T\to\infty}Q_{T}\big{(}A^{(>)}_{T,r_{2}(T)}\big{)}&=0.\end{split}$

Our main result is stated in the following theorem.

######  Theorem 1.1.

There exists positive constants $C_{d},c_{d}$ and $C$ not depending on $N,T$
and $\beta$ such that,

  * (i)

for $d=1$, for all $\beta,N\geq 1$, and $T\geq c_{1}^{-1}$,

$Q_{T}\left(c_{1}\beta^{\frac{1}{3}}N^{\frac{1}{3}}T\leq R_{T}\leq
C_{1}\beta^{\frac{1}{3}}N^{\frac{1}{3}}T\right)\geq
1-\exp\left(-C\beta^{\frac{2}{3}}N^{\frac{5}{3}}T\right),$

  * (ii)

for $d=2$, for all $\beta\geq 1$, $N\geq(c_{2}^{-4/3}\vee 1)$ and
$c_{2}^{-4/3}\leq T\leq N$,

$Q_{T}\left(c_{2}\beta^{\frac{1}{4}}N^{\frac{1}{4}}T^{\frac{3}{4}}(\log(\beta
T))^{-\frac{1}{2}}\leq R_{T}\leq
C_{2}\beta^{\frac{1}{4}}N^{\frac{1}{4}}T^{\frac{3}{4}}(\log(\beta
T))^{\frac{1}{2}}\right)\\\ \geq
1-\exp\left(-C\beta^{\frac{1}{2}}N^{\frac{3}{2}}T^{\frac{1}{2}}\log(\beta
T)\right),$

  * (iii)

for $d=3$, for all $\beta\geq 1$, $N\geq(c_{3}^{-2}\vee 1)$ and
$c_{3}^{-2}\leq T\leq N$,

$Q_{T}\left(c_{3}\beta^{\frac{1}{6}}N^{\frac{1}{6}}T^{\frac{1}{2}}(\log(\beta
T))^{-\frac{1}{3}}\leq R_{T}\leq
C_{3}\beta^{\frac{1}{4}}N^{\frac{1}{4}}T^{\frac{3}{4}}(\log(\beta
T))^{\frac{1}{2}}\right)\\\ \geq
1-\exp\left(-C\beta^{\frac{1}{2}}N^{\frac{3}{2}}T^{\frac{1}{2}}\log(\beta
T)\right).$

Note that the upper and lower bounds in part (iii) of include the physical
result given in (1.2).

The proof of Theorem 1.1 is given in Section 3.

##  2\. strategy

To analyze (1.5), suppose we are given functions
$r_{i}:(0,\infty)\to(0,\infty)$ with $i\in\\{1,2\\}$. We define

(2.1)
$\begin{split}q_{T}^{(<)}&=Q_{T}(A^{(<)}_{T,r_{1}(T)})=\frac{1}{Z_{T}}E^{P_{T}}\Big{[}\mathbf{1}_{A^{(<)}_{T,r_{1}(T)}}\mathcal{E}_{T}\Big{]},\\\
q_{T}^{(>)}&=Q_{T}(A^{(>)}_{T,r_{2}(T)})=\frac{1}{Z_{T}}E^{P_{T}}\Big{[}\mathbf{1}_{A^{(>)}_{T,r_{2}(T)}}\mathcal{E}_{T}\Big{]}.\end{split}$

Note that $q_{T}^{(<)},q_{T}^{(>)}$ implicitly depend on $r_{1},r_{2}$ and
also $d,N,\beta$. Then

(2.2) $\displaystyle q_{T}^{(<)}$
$\displaystyle\leq\frac{1}{Z_{T}}\sup_{\omega\in
A^{(<)}_{T,r_{1}(T)}}\mathcal{E}_{T}(\omega),$ (2.3) $\displaystyle
q_{T}^{(>)}$
$\displaystyle\leq\frac{1}{Z_{T}}E^{P_{T}}\Big{[}\mathbf{1}_{A^{(>)}_{T,r_{2}(T)}}\Big{]}=\frac{1}{Z_{T}}P_{T}\big{(}A^{(>)}_{T,r_{2}(T)}\big{)}.$

We first consider (2.2). Now (1.6) shows that on $A^{(<)}_{T,r_{1}(T)}$, at
least $[N/2]$ Brownian motions satisfy $\sup_{t\in[0,T]}|B^{(k)}_{t}|\leq
r_{1}(T)$. Here for any $x\in\mathbb{R}$, $[x]$ is the greatest integer less
than or equal to $x$. On $A^{(<)}_{T,r_{1}(T)}$, let
$\\{k_{1},\dots,k_{[N/2]}\\}$ be the first $[N/2]$ indices of the Brownian
motions satisfying this condition, and define $L_{T}^{(\text{med})}$ be the
total occupation measure of these Brownian motions.

On $A^{(<)}_{T,r_{1}(T)}$ we have that $L^{(\text{med})}_{T}(\cdot)$ is
supported on $\mathbf{B}_{r_{1}(T)+1}(0)$ and
$\int_{\mathbb{R}^{d}}L^{(\text{med})}_{T}(x)dx=[N/2]T$. The Cauchy-Schwarz
inequality shows that among nonnegative functions $f$ supported on
$\mathbf{B}_{r_{1}(T)+1}(0)$, such that $\int_{\mathbb{R}^{d}}f(x)dx=[N/2]T$,
the minimum of $\int_{\mathbb{R}^{d}}f(x)^{2}dx$ is achieved when $f$ equals a
constant $K$ on $\mathbf{B}_{r_{1}(T)+1}(0)$, and in that case

$K=\frac{[N/2]T}{V_{d}\cdot(r_{1}(T)+1)^{d}},$

where $V_{d}$ is the volume the unit $d$-dimensional ball.

So, on $A^{(<)}_{T,r_{1}}$ we have

$\begin{split}\int_{\mathbb{R}^{d}}L_{T}(y)^{2}dy&\geq
K^{2}|\mathbf{B}_{r_{1}(T)+1}(0)|\\\
&\geq\left(\frac{[N/2]T}{V_{d}\cdot(r_{1}(T)+1)^{d}}\right)^{2}V_{d}\cdot(r_{1}(T)+1)^{d}\\\
&=C\frac{N^{2}T^{2}}{(r_{1}(T)+1)^{d}}.\end{split}$

Then by the definition of $\mathcal{E}_{T}$, we have

(2.4) $q_{T}^{(<)}\leq\frac{1}{Z_{T}}\sup_{\omega\in
A^{(<)}_{T,r_{1}(T)}}\mathcal{E}_{T}(\omega)\leq\frac{1}{Z_{T}}\exp\left(-\beta
C\frac{N^{2}T^{2}}{(r_{1}(T)+1)^{d}}\right).$

Now we turn to (2.3). In order to bound $q_{T}^{(>)}$, we use the following
large deviations result.

######  Lemma 2.1.

Given $p\in(0,1)$, let $(X_{i})_{i\geq 1}$ be an infinite sequence of
Bernoulli random variables with $P(X_{i}=1)=p$ and $P(X_{i}=0)=q:=1-p$. Define
$S_{n}:=\sum_{i=1}^{n}X_{i}$. For $\alpha\in(p,1)$ we have

$P(S_{n}>\alpha n)\leq\left(\frac{(1-\alpha)p}{\alpha q}\right)^{\alpha
n}\left[q+p\frac{\alpha q}{(1-\alpha)p}\right]^{n}.$

###### Proof of Lemma 2.1.

Such large deviations results are well-known, but they are usually expressed
in asymptotic form. We need an upper bound for the probability, and we give a
self-contained proof.

Using the moment generating function of the binomial distribution and choosing
$t>0$, for $\alpha\in(p,1)$ we get

(2.5) $\begin{split}P\left(S_{n}>\alpha
n\right)&=P\big{(}\exp\left(tS_{n}\right)>\exp\left(\alpha tn\right)\big{)}\\\
&\leq\frac{E\left[\exp\left(tS_{n}\right)\right]}{\exp\left(\alpha
tn\right)}=\exp\left(-\alpha tn\right)\left[q+pe^{t}\right]^{n}\\\
&=\left(e^{-\alpha t}\left[q+pe^{t}\right]\right)^{n}.\end{split}$

Let $f(t)=e^{-\alpha t}[q+pe^{t}]$ and note that $f$ is differentiable on
$\mathbb{R}$. We compute

$\begin{split}f^{\prime}(t)&=-\alpha e^{-\alpha t}[q+pe^{t}]+e^{-\alpha
t}pe^{t}\\\ &=e^{-\alpha t}[-\alpha q+(1-\alpha)pe^{t}]\end{split}$

We find that $f(t)\to\infty$ as $t\to\infty$. Since $\alpha>p$, we have

(2.6) $f^{\prime}(0)=-\alpha q+(1-\alpha)p=p-\alpha(p+q)=p-\alpha<0.$

Thus $f$ achieves its minimum over $[0,\infty)$ in $(0,\infty)$. Let $t^{*}$
be the infimum of those $t\in(0,\infty)$ for which the minimum is achieved.

Solving $f^{\prime}(t^{*})=0$, we find $-\alpha q+(1-\alpha)pe^{t^{*}}=0$ and
so

$t^{*}=\log\frac{\alpha q}{(1-\alpha)p}.$

Combining (2.5) and (2.6) and substituting $t=t^{*}$ gives

$P(S_{n}>\alpha n)\leq\left(\frac{(1-\alpha)p}{\alpha q}\right)^{\alpha
n}\left[q+p\frac{\alpha q}{(1-\alpha)p}\right]^{n}.$

This proves Lemma 2.1. ∎

Now let $r=r_{2}(T)$, and define

$X_{i}=X_{i}^{(T)}:=\mathbf{1}_{\\{\sup_{t\in[0,T]}|B^{(k)}_{t}|>r_{2}(T)\\}},$

and $p=p_{T}:=P(X_{i}^{(T)}=1)$.

Assuming that $p_{T}\in(0,1/2)$ and $\alpha=1/2$, we conclude

(2.7)
$\begin{split}q_{T}^{(>)}&=\frac{1}{Z_{T}}P(S_{N}>N/2)\leq\frac{1}{Z_{T}}\left(\frac{p_{T}}{1-p_{T}}\right)^{N/2}\left[2(1-p_{T})\right]^{N}\\\
&=\frac{1}{Z_{T}}2^{N}[p_{T}(1-p_{T})]^{N/2}\leq\frac{1}{Z_{T}}(4p_{T})^{N/2}\end{split}$

We expect that $p_{T}$ is close to 0 for large $T$, so not much is lost in the
final step above.

The probability that a single Brownian motion exits $\mathbf{B}_{r_{2}(T)}(0)$
by time $T$ is bounded by

(2.8) $p_{T}=P_{T}\Big{(}\sup_{t\in[0,T]}|B^{(k)}_{t}|>r_{2}(T)\Big{)}\leq
C\frac{1}{r_{2}(T)}\exp\left(-\frac{r_{2}(T)^{2}}{2T}\right),$

by standard Brownian estimates.

Continuing, we use (2.7) and (2.8) to get

$q_{T}^{(>)}\leq\frac{1}{Z_{T}}(4p_{T})^{N/2}\leq\frac{1}{Z_{T}}\left(\frac{C}{r_{2}(T)}\right)^{N/2}\exp\left(-\frac{Nr_{2}(T)^{2}}{2T}\right).$

Now assuming that $r_{2}(T)^{2}/T>C_{0}>0$ we can absorb $(C/r_{2}(T))^{N/2}$
into the exponential to get

(2.9)
$q_{T}^{(>)}\leq\frac{1}{Z_{T}}\exp\left(-\frac{CNr_{2}(T)^{2}}{T}\right),$

for some constant $C>0$.

From (1.6) it follows that on $A^{(>)}_{T,r_{2}}$, at least $[N/2]$ of the
Brownian paths exit from $\mathbf{B}_{r_{2}(T)}(0)$ within time $[0,T]$.

Our next argument is only heuristic, but it allows us to guess a formula for
$R_{T}$. Recall that $Q_{T}$ in (1.5) involves the ratio of $P_{T}$ to
$Z_{T}$. We think of $R_{T}$ as the critical value of $r_{1}(T)$ and
$r_{2}(T)$ for which both $q^{(<)}_{T},q^{(>)}_{T}$ are close to $Z_{T}$. We
believe that if $r_{1}(T)\approx r_{2}(T)\approx R_{T}$, then
$q^{(<)}_{T},q^{(>)}_{T}$ will be close. Thus we set $r_{1}(T)\approx
r_{2}(T)$ and equalize the powers appearing in (2.4) and (2.9). Following this
path, we conclude that (ignoring constants)

$\frac{Nr^{2}_{2}(T)}{T}\approx\beta\frac{N^{2}T^{2}}{r_{1}(T)^{d}}$

hence setting $r_{1}(T)\approx r_{2}(T)\approx R_{T}$, we get the guess

$R_{T}\approx\beta^{\frac{1}{d+2}}N^{\frac{1}{d+2}}T^{\frac{3}{d+2}}.$

This leads us to guess that, up to logarithmic factors,

(2.10)
$Z_{T}\approx\beta^{\frac{2}{d+2}}N^{\frac{d+4}{d+2}}T^{\frac{4-d}{d+2}}.$

We describe our results regarding $Z_{T}$ in the following theorem.

######  Theorem 2.2.

Let $Z_{T}$ be defined as in (1.5). Then there exists a constant
$C_{\ref{prop-z}}>0$ not depending on $N,T$ and $\beta$ such that,

  * (i)

for $d=1$, for all $\beta,T\geq 1$,

$\log Z_{T}\geq-C_{\ref{prop-z}}\beta^{\frac{2}{3}}N^{\frac{5}{3}}T,$

  * (ii)

for $d=2,3$, for all $\beta\geq 1$ and $1\leq T\leq N$,

$\log Z_{T}\geq-
C_{\ref{prop-z}}\beta^{\frac{1}{2}}N^{\frac{3}{2}}T^{\frac{1}{2}}\log(\beta
T),$

The proof of Theorem 2.2 is given in Section 3.

##  3\. Proof of Theorems 1.1 and 2.2

###### Proof of Theorem 2.2.

In order to derive the bounds Theorem 2.2 we use a change of measure that will
impose a radial drift $\lambda_{k}(t)$ on each Brownian motion $B^{(k)}$, of
magnitude

(3.1) $|\lambda_{k}(t)|=\kappa(\alpha+1)t^{\alpha},$

for some $\kappa>0$ and $\alpha<0$ to be determined. Now we describe the
directions $(\theta_{k})_{k\in\\{1,\dots,N\\}}$ of the drifts, where each
$\theta_{k}$ is a point on the unit sphere
$\mathbf{S}_{d}\subset\mathbb{R}^{d}$. Let
$\theta:=(\theta_{k})_{k\in\\{1,\dots,N\\}}$ be an ensemble of i.i.d. random
points chosen according to the uniform probability measure on
$\mathbf{S}_{d}$. Specifically, given $T,N$ we define
$(\theta_{k})_{k\in\\{1,\dots,N\\}}$ over a probability space
$(\Omega_{\theta},\mathcal{F}_{\theta},P_{\theta})$ and form the product space

$(\overline{\Omega},\mathcal{G},\mathbb{P})=(\Omega_{\theta}\times\Omega,\mathcal{F}_{\theta}\times\mathcal{F},P_{\theta}\times
P_{T}).$

Sometimes we write $\mathbb{P}_{T}$ to emphasize the dependence on $T$. Note
that there is also an implicit dependence on $N$. Finally, we denote
$\lambda:=(\lambda_{k})_{k\in\\{1,\dots,N\\}}$.

Roughly speaking, we want the drift to be stronger than the standard deviation
$t^{1/2}$ of Brownian motion. From (3.1) it follows that the accumulated drift
up to time $t$ is given by

(3.2) $\kappa(\alpha+1)\int_{0}^{t}s^{\alpha}ds=\kappa t^{\alpha+1},$

so we require that $\alpha>-1/2$.

Given $\theta$ and $N,T$, we denote the measure on canonical path space
$(C[0,T])^{N}$ which is induced by the drifts $\lambda$ as $P^{\lambda}_{T}$.
The Radon-Nikodym derivative with respect to $P_{T}$ in (1.5) is given by

$\frac{dP^{\lambda}_{T}}{dP_{T}}=\exp\left(\sum_{k=1}^{N}\left\\{\int_{0}^{T}\lambda_{k}(t)\cdot
dB^{(k)}_{t}-\frac{1}{2}\int_{0}^{T}|\lambda_{k}(t)|^{2}dt\right\\}\right).$

We can also define the corresponding product probability
$\mathbb{P}^{\lambda}_{T}$ as before, using $P^{\lambda}_{T}$ instead of
$P_{T}$.

We can express $Z_{T}$ in (1.5) in terms of $\mathcal{E}_{T}$ and the Radon-
Nikodym derivative as follows

$Z_{T}=E^{\mathbb{P}_{T}^{\lambda}}\left[\mathcal{E}_{T}\left(\frac{dP^{\lambda}_{T}}{dP_{T}}\right)^{-1}\right].$

We can use Jensen’s inequality on $\log Z_{T}$, since the logarithm function
is concave to get

(3.3) $\begin{split}\log Z_{T}&\geq
E^{\mathbb{P}_{T}^{\lambda}}\left[\log\mathcal{E}_{T}-\log\left(\frac{dP^{\lambda}_{T}}{dP_{T}}\right)\right]\\\
&\geq-\beta
E^{\mathbb{P}_{T}^{\lambda}}\left[\int_{\mathbb{R}^{d}}L_{T}(y)^{2}dy\right]-E^{\mathbb{P}_{T}^{\lambda}}\left[\log\left(\frac{dP^{\lambda}_{T}}{dP_{T}}\right)\right]\\\
&=:-\beta I_{1}(d,\beta,N,T)-I_{2}(d,\beta,N,T).\end{split}$

Using (3.1) we can easily compute

(3.4)
$\begin{split}I_{2}(d,\beta,N,T)&=E^{\mathbb{P}_{T}^{\lambda}}\left[\sum_{k=1}^{N}\left\\{\int_{0}^{T}\lambda_{k}(t)\cdot
dB^{(k)}_{t}-\frac{1}{2}\int_{0}^{T}|\lambda_{k}(t)|^{2}dt\right\\}\right]\\\
&=-\frac{N}{2}\kappa^{2}(\alpha+1)^{2}E\left[\int_{0}^{T}t^{2\alpha}dt\right]\\\
&=-\frac{\kappa^{2}(\alpha+1)^{2}}{2}N\cdot\frac{T^{2\alpha+1}}{2\alpha+1}.\end{split}$

We can compare $I_{2}(d,\beta,N,T)$ in (3.4) with (2.10) in order to determine
the constants in the drift. Neglecting multiplicative constants it follows
that we must have

$\kappa^{2}N\cdot\frac{T^{2\alpha+1}}{2\alpha+1}=\beta^{\frac{2}{d+2}}N^{\frac{d+4}{d+2}}T^{\frac{4-d}{d+2}}.$

This will lead to the following choice of drift parameters:

(3.5)
$\kappa=\beta^{\frac{1}{d+2}}N^{\frac{1}{d+2}},\quad\alpha=-\frac{d-1}{d+2},\quad
d=1,2,3.$

######  Remark 3.1.

While it is tempting to use the drift in (3.5) for $d=1,2,3$, we find that in
the case of $d=3$ it is suboptimal. This choice of drift for $d=3$ yields the
following bound on $Z_{T}$.

(3.6) $\log Z_{T}\geq-
C_{\ref{prop-z}}\beta^{\frac{3}{5}}N^{\frac{8}{5}}T^{\frac{4}{5}}\log(\beta
T).$

Since (3.6) is suboptimal we do not prove it, but the interested reader can
verify it by modifying the arguments in Sections 4, 5, and 6.

It is better to use the same drift parameter for $d=3$ as for $d=2$, namely
$\kappa=\beta^{\frac{1}{4}}N^{\frac{1}{4}}$, $\alpha=-1/4$. This choice gives
the bound in Theorem 2.2(ii), that is for $d=3$,

$\log Z_{T}\geq-
C_{\ref{prop-z}}\beta^{\frac{1}{2}}N^{\frac{3}{2}}T^{\frac{1}{2}}\log(\beta
T),$

which is clearly better than (3.6) for $\beta,N,T\geq 1$. The reason for that
is that the bound on $Z_{T}$ depends on a comparison between the two
components on the right hand side of (3.3): $I_{1}(d,\beta,N,T)$ which
involves the occupation measure $L_{T}$, and on $I_{2}(d,\beta,N,T)$ which
contains the Radon-Nikodym derivative and is computed in (3.4). The bound on
$I_{1}(d,\beta,N,T)$ in Proposition 3.2 below, is obtained by bounding the
terms in right-hand side of (4.2): $J_{1}(d,\beta,N,T)$ which depends on self-
intersection occupation measure of each branch of the polymer and
$J_{2}(d,\beta,N,T)$ which involves cross-intersection occupation measure of
pairs of branches. The bound on $J_{1}(d,\beta,N,T)$, which is derived on
Section 6, is the more restrictive one and eventually determines the result of
Proposition 3.2. In particular in the proof of Lemma 6.1 (see (6.7)) we show
that this bound should be similar to $d=2,3$. Hence choosing similar drifts
for the cases of $d=2,3$ leads to an optimal bound for our method.

For the reminder of the paper, we therefore fix

(3.7) $\kappa=\begin{cases}\beta^{\frac{1}{3}}N^{\frac{1}{3}},&d=1,\\\
\beta^{\frac{1}{4}}N^{\frac{1}{4}},&d=2,3,\\\
\end{cases}\qquad\alpha=\begin{cases}0,&\ d=1,\\\ -\frac{1}{4},&\
d=2,3.\end{cases}$

The rest of this paper is dedicated to the estimation of $I_{1}(d,\beta,N,T)$.
This is essentially given in the following proposition which together with
(3.3) and (3.4) concludes the proof of Theorem 2.2.

######  Proposition 3.2.

There exists a constant $C_{\ref{prop-i-2}}>0$ not depending on $N,T$ and
$\beta$ such that,

  * (i)

for $d=1$, for all $\beta,T\geq 1$,

$I_{1}(1,\beta,N,T)\leq
C_{\ref{prop-i-2}}\beta^{-\frac{1}{3}}N^{\frac{5}{3}}T.$

  * (ii)

for $d=2,3$, for all $\beta\geq 1$ and $1\leq T\leq N$,

$I_{1}(d,\beta,N,T)\leq
C_{\ref{prop-i-2}}\beta^{-\frac{1}{2}}N^{\frac{3}{2}}T^{\frac{1}{2}}\log(\beta
T).$

The proof of Proposition 3.2 for $d=2,3$ is postponed to Section 4 and the
case of $d=1$, which is much simpler, is postponed to Section 7. ∎

###### Proof of Theorem 1.1.

Note that Theorem 2.2 provides a lower bound on $Z_{T}$ of the form

(3.8) $\log Z_{T}\geq-f(d,\beta,N,T),$

where

$f(d,\beta,N,T)=\begin{cases}C_{2.2}\beta^{2/3}N^{5/3}T&\text{if $d=1$},\\\
C_{2.2}\beta^{1/2}N^{3/2}T^{1/2}&\text{if $d=2,3$}.\end{cases}$

From (1.5), (1.7), (2.1), (2.4) and (3.8) we have

(3.9) $\displaystyle\log Q_{T}\big{(}A^{(<)}_{T,r_{1}(T)}\big{)}$
$\displaystyle=\log q_{T}^{(<)}-\log Z_{T}$ $\displaystyle\leq-\beta
C\frac{N^{2}T^{2}}{(1+r_{1}(T))^{d}}+f(d,\beta,N,T),$

hence by choosing $r_{1}(T)$ as in the upper bound in the statement of Theorem
1.1, with $c_{d}$ small enough and $\beta,N,T$ as in the hypothesis, we ensure
that $r_{1}(T)\geq 1$ and therefore,

$\displaystyle-\beta C\frac{N^{2}T^{2}}{(1+r_{1}(T))^{d}}+f(d,\beta,N,T)$
$\displaystyle\leq-\beta\hat{C}\frac{N^{2}T^{2}}{r_{1}(T)^{d}}+f(d,\beta,N,T)$
$\displaystyle<-cf(d,\beta,N,T),$

for some constant $c>0$ and we get the lower bound on $R_{T}$.

Similarly from (1.5), (1.7), (2.1), (2.9) and (3.8) we have

(3.10) $\displaystyle\log Q_{T}\big{(}A^{(>)}_{T,r_{2}(T)}\big{)}$
$\displaystyle=\log q_{T}^{(>)}-\log Z_{T}$
$\displaystyle\leq-\frac{CNr_{2}(T)^{2}}{T}+f(d,\beta,N,T),$

hence by choosing $r_{2}(T)$ as in the statement of Theorem 1.1 we ensure that

$-\frac{CNr_{2}(T)^{2}}{T}+f(d,\beta,N,T)<-c^{\prime}f(d,\beta,N,T),$

for some constant $c^{\prime}>0$ and we get the upper bound on $R_{T}$. Note
for $d=1$ that $r_{1}(T)$ and $r_{2}(T)$ agree up to a constant; for $d=2$,
$r_{1}(T)$ and $r_{2}(T)$ agree up to logarithmic terms; while in $d=3$ there
is a gap between them. ∎

##  4\. Proof of Proposition 3.2 for $d=2,3$

This section is dedicated to the bound on $I_{1}(d,\beta,N,T)$ in Proposition
3.2 for $d=2,3$.

###### Proof of Proposition 3.2 for $d=2,3$.

Recall that $L_{T}$ was defined in (1.4). From (3.3) and the statement of
Proposition 3.2, we see that we need the bound for $d=2,3$,

(4.1)
$E^{\mathbb{P}_{T}^{\lambda}}\left[\int_{\mathbb{R}^{d}}L_{T}(y)^{2}dy\right]\leq\beta^{-\frac{1}{2}}N^{\frac{3}{2}}T^{\frac{1}{2}}.$

for $\alpha$ and $\kappa$ as in (3.7).

Note that

$\begin{split}\int_{\mathbb{R}^{d}}L_{T}(y)^{2}dy&=\int_{\mathbb{R}^{d}}\left(\sum_{k=1}^{N}\int_{0}^{T}\mathbf{1}_{\mathbf{B}_{1}(y)}(B_{t}^{(k)})dt\right)^{2}dy\\\
&=\sum_{k,\ell=1}^{N}\int_{0}^{T}\int_{0}^{T}\left(\int_{\mathbb{R}^{d}}\mathbf{1}_{\mathbf{B}_{1}(y)}(B_{t}^{(k)})\mathbf{1}_{\mathbf{B}_{1}(y)}(B_{s}^{(\ell)})dy\right)dtds.\end{split}$

In fact, for $a,b\in\mathbb{R}^{d}$, we have

$\int_{\mathbb{R}^{d}}\mathbf{1}_{\mathbf{B}_{1}(y)}(a)\mathbf{1}_{\mathbf{B}_{1}(y)}(b)dy\leq\mathbf{1}_{\mathbf{B}_{2}(0)}(b-a)|\mathbf{B}_{1}(0)|=C_{d}\mathbf{1}_{\mathbf{B}_{2}(0)}(b-a).$

Thus, if $f^{(k,\ell,\alpha)}_{t,s}(z)$ is the probability density function of
$B^{(k)}_{t}-B^{(\ell)}_{s}$ under $\mathbb{P}_{T}^{\lambda}$, then using
Fubini’s Theorem we get,

(4.2)
$\displaystyle\int_{\mathbb{R}^{d}}E^{\mathbb{P}_{T}^{\lambda}}\left[L_{T}(y)^{2}\right]dy$
$\displaystyle\leq
C_{d}\sum_{k,\ell=1}^{N}\iint_{[0,T]^{2}}\int_{\mathbf{B}_{2}(0)}f^{(k,\ell,\alpha)}_{t,s}(z)dzdsdt$
$\displaystyle=C_{d}\sum_{k=1}^{N}\iint_{[0,T]^{2}}\int_{\mathbf{B}_{2}(0)}f^{(k,k,\alpha)}_{t,s}(z)dzdsdt$
$\displaystyle\quad+C_{d}\sum_{k\not=\ell}\iint_{[0,T]^{2}}\int_{\mathbf{B}_{2}(0)}f^{(k,\ell,\alpha)}_{t,s}(z)dzdsdt$
$\displaystyle=:J_{1}(d,\beta,N,T)+J_{2}(d,\beta,N,T).$

Note that $J_{1}(d,\beta,N,T)$ represents the represents the sum of mean-
squared self-intersection occupation measure of each branch of the star
polymer. $J_{2}(d,\beta,N,T)$ represents the sum over all pairs of branches of
their mean-squared cross-intersection occupation measure. The following
proposition gives some essential bounds on each of these terms.

######  Proposition 4.1.

For $d=2,3$ there exists a constant $C>0$ such that for all $1\leq T\leq N$,
$N\in\mathbb{Z}^{+}$ and $\beta\geq 1$ the following bound holds:

$J_{i}(d,\beta,N,T)\leq
C\beta^{-\frac{1}{2}}N^{\frac{3}{2}}T^{\frac{1}{2}}\log(\beta T),\quad i=1,2.$

The proof of Proposition 4.1 is long and involved. In Section 5 we prove the
bound on cross-intersection occupation measure, which involves two different
Brownian motions $B^{(k_{i})}$ for $i=1,2$ and $k_{1}\neq k_{2}$, . The bound
on self-intersection occupation measure is derived in Section 6. From (4.1),
(4.2) and Proposition 4.1 we get the bounds in Proposition 3.2. ∎

##  5\. Cross-intersections occupation measure for $d=2,3$

In this section we derive the bounds on $J_{2}(d,\beta,N,T)$ from Proposition
4.1 for $d=2,3$. We recall that $f^{(k,\ell,\alpha)}_{t,s}(z)$ is the
probability density function of $B^{(k)}_{t}-B^{(\ell)}_{s}$ under
$\mathbb{P}_{T}^{\lambda}$. Our next task is to estimate
$f^{(k,\ell,\alpha)}_{t,s}(z)$. Since $f$ only depends on the angle $\theta$
between the two drifts of $B^{(k)}$ and $B^{(\ell)}$, from now on we write
$f^{\theta,\alpha}_{t,s}$ for our probability density.

Instead of working with $P^{\lambda}_{T}$, we will work with $P_{T}$ and
consider processes $B^{(k)}_{t}+D^{(k)}_{t},B^{(\ell)}_{s}+D^{\ell}_{s}$,
where $D^{(k)},D^{(\ell)}$ are the respective drift processes. First note that

$B^{(k)}_{t}-B^{(\ell)}_{s}\sim\mathcal{N}(0,t+s),$

where $\mathcal{N}(\cdot,\cdot)$ stands for the $d$-dimensional normal
distribution.

Recall that the drift magnitudes are given by (3.2), namely

(5.1) $\displaystyle|D^{(k)}_{t}|=\kappa t^{\alpha+1},$
$\displaystyle|D^{(\ell)}_{s}|=\kappa s^{\alpha+1}.$

For the remainder of the proof we assume that $\theta\in[0,\pi]$. First we get
a lower bound on $|z-D^{(k)}_{t}+D^{(\ell)}_{s}|^{2}$. Let
$a=D^{(k)}_{t}-D^{(\ell)}_{s}$. Then

$\begin{split}|a|^{2}=|(a-z)+z|^{2}\leq 2|z-a|^{2}+2|z|^{2}\leq
2|z-a|^{2}+8,\end{split}$

since $|z|\leq 2$ in the domain of integration of $J_{2}(d,\beta,N,T)$ (see
(4.2)). Now we use the law of cosines to deduce

(5.2)
$\begin{split}|D^{(k)}_{t}-D^{(\ell)}_{s}|^{2}&=|D^{(k)}_{t}|^{2}+|D^{(\ell)}_{s}|^{2}-2|D^{(k)}_{t}|\cdot|D^{(\ell)}_{s}|\cos\theta\\\
&=\left(|D^{(k)}_{t}|-|D^{(\ell)}_{s}|\right)^{2}+2|D^{(k)}_{t}|\cdot|D^{(\ell)}_{s}|\left[1-\cos\theta\right].\end{split}$

Elementary geometry shows that for some constant $C>0$ and for all
$\theta\in[0,\pi]$,

(5.3) $1-\cos\theta\geq C\theta^{2}$

and so

(5.4)
$|D^{(k)}_{t}-D^{(\ell)}_{s}|^{2}\geq\left(|D^{(k)}_{t}|-|D^{(\ell)}_{s}|\right)^{2}+C|D^{(k)}_{t}|\cdot|D^{(\ell)}_{s}|\theta^{2}.$

We also recall the following integral bound. For any $K>0$ we have

(5.5)
$\int_{0}^{\infty}\exp\left(-K\theta^{2}\right)d\theta=\frac{\sqrt{\pi}}{2\sqrt{K}}.$

######  Remark 5.1.

The bound in (5.5) will be used in Lemma 5.2 (see (5.11)) and in Lemma 5.4
(see (5.29)) in order to bound $J_{2}(d,\beta,N,T)$ for two different cases.
Note that the aforementioned bounds can be improved for $d=3$ by using the
following equality for any $K>0$,

$\int_{0}^{\infty}\exp\left(-K\theta^{2}\right)\theta d\theta=\frac{1}{2K},$

for the expression for $J_{2}(d,\beta,N,T)$ in (5.7) which has an extra factor
of $\theta$ (see explanation after (5.7)). As mentioned in Remark 3.1 this
will not have any affect on the result of Theorem 2.2. Therefore, for the sake
of simplicity we provide one proof for the bounds $J_{2}(d,\beta,N,T)$ from
Proposition 4.1 for $d=2,3$ using (5.5).

We derive a preliminary bound for $J_{2}(d,\beta,N,T)$ in (4.2) where we
distinguish between the following cases.

Case 1: $d=2$. Note that $J_{2}(2,\beta,N,T)$ is bounded by

(5.6) $\displaystyle J_{2}(2,\beta,N,T)$ $\displaystyle\leq
CN^{2}\int_{0}^{\pi}\iint_{0\leq s\leq t\leq T}\frac{1}{t+s}$
$\displaystyle\qquad\times\int_{\mathbf{B}_{2}(0)}\exp\left(-\frac{\big{|}z-D_{t}^{(k)}+D_{s}^{(\ell)}\big{|}^{2}}{2(t+s)}\right)dzdsdtd\theta.$

Case 2: $d=3$. By the same argument as in Case 1 we get,

(5.7) $\displaystyle J_{2}(3,\beta,N,T)$ $\displaystyle\leq
CN^{2}\int_{0}^{\pi}\iint_{0\leq s\leq t\leq T}\frac{1}{(t+s)^{3/2}}$
$\displaystyle\qquad\times\int_{\mathbf{B}_{2}(0)}\exp\left(-\frac{\big{|}z-D_{t}^{(k)}+D_{s}^{(\ell)}\big{|}^{2}}{2(t+s)}\right)dzdsdtd\theta.$

Note that in the second integral above, using polar coordinates would give
$\theta d\theta$. The interested reader can check that using $d\theta$ gives
the same end result. Of course we can bound $\theta$ by $\pi$ and so replace
$\theta d\theta$ by $d\theta$.

Let $V$ be the subspace generated by the first two coordinates of
$\mathbb{R}^{3}$, so for $z=(z_{1},z_{2},z_{3})$ we denote by
$z_{V}=(z_{1},z_{2})$. In fact we can find a coordinate system in which
$D_{t}^{(k)}$ is parallel to $z_{1}$, so the above integral is bounded by

(5.8) $\displaystyle J_{2}(3,\beta,N,T)$ $\displaystyle\leq
CN^{2}\int_{0}^{\pi}\iint_{0\leq s\leq t\leq
T}\frac{1}{(t+s)}\int_{{\mathbf{B}}^{V}_{2}(0)}\exp\left(-\frac{\big{|}z_{V}-D_{t}^{(k)}+D_{s}^{(\ell)}\big{|}^{2}}{2(t+s)}\right)dz_{V}$
$\displaystyle\qquad\qquad\qquad\quad\qquad\qquad\times\frac{1}{(t+s)^{1/2}}\int_{-\infty}^{\infty}\exp\left(-\frac{z_{3}^{2}}{2(t+s)}\right)dz_{3}dsdtd\theta$
$\displaystyle\leq CN^{2}\int_{0}^{\pi}\iint_{0\leq s\leq t\leq
T}\frac{1}{(t+s)}$
$\displaystyle\qquad\qquad\quad\qquad\qquad\times\int_{{\mathbf{B}}^{V}_{2}(0)}\exp\left(-\frac{\big{|}z_{V}-D_{t}^{(k)}+D_{s}^{(\ell)}\big{|}^{2}}{2(t+s)}\right)dz_{V}dsdtd\theta,$

where ${\mathbf{B}}^{V}_{2}(0)$ is the projection of ${\mathbf{B}}_{2}(0)$ to
$V$. From (5.6) and (5.8) it follows that we need to bound the same integral
for the cases where $d=2,3$ but $D_{t}^{(k)},D_{s}^{(\ell)}$ will change
according to the dimension as implied by (3.7) and (5.1). In the following we
therefore work on to the combined cases $d=2,3$.

In order to do that we distinguish between a few cases which depend on the
range of $(t,s)$ in the above integral.

Define

(5.9) $\displaystyle\mathbf{R}_{1}$ $\displaystyle=\left\\{(s,t):0\leq s\leq
t\leq T,t\geq 4\right\\},$ $\displaystyle\mathbf{R}_{2}$
$\displaystyle=\left\\{(s,t):0\leq s\leq t\leq
T,\,t<4,\,|D_{t}^{(0)}-D_{s}^{(\ell)}|^{2}\leq 8\right\\},$
$\displaystyle\mathbf{R}_{3}$ $\displaystyle=\left\\{(s,t):0\leq s\leq t\leq
T,\,t<4,\,|D_{t}^{(0)}-D_{s}^{(\ell)}|^{2}>8\right\\}.$

Moreover for $i=1,2,3,$ and $d=2,3$ define:

(5.10) $\displaystyle\overline{J}(d,\beta,N,T,\mathbf{R}_{i})$
$\displaystyle=CN^{2}\int_{0}^{\pi}\iint_{\mathbf{R}_{i}}\frac{1}{t+s}$
$\displaystyle\times\int_{\mathbf{B}_{2}(0)}\exp\left(-\frac{\big{|}z-D_{t}^{(0)}+D_{s}^{(\ell)}\big{|}^{2}}{2(t+s)}\right)dzdsdtd\theta.$

We now present a sequence of technical lemmas that will help us to bound
$J_{2}(d,\beta,N,T)$. The proof of Proposition 4.1 for $J_{2}(d,\beta,N,T)$ is
given in the end of this section.

######  Lemma 5.2.

For $d=2,3$, there exists a constant $C_{\ref{lem-j-r1}}>0$ such that for all
$\beta,T\geq 1$ and $N\in\mathbb{Z}^{+}$ the following bound holds:

$\overline{J}(d,\beta,N,T,\mathbf{R}_{1})\leq C_{\ref{lem-
j-r1}}\beta^{-1/2}N^{3/2}T^{1/2}.$

###### Proof.

Using (5.4) we get

$|z-D_{t}^{(k)}+D_{s}^{(\ell)}|^{2}\geq\left(|D^{(k)}_{t}|-|D^{(\ell)}_{s}|\right)^{2}+C|D^{(k)}_{t}|\cdot|D^{(\ell)}_{s}|\theta^{2}-4,$

since $|z|\leq 2$ in the domain of integration of
$\overline{J}(d,\beta,N,T,\mathbf{R}_{i})$ (see (5.10)).

Note that since on $\mathbf{R}_{1}$ we have $s+t>4$, it follows that
$4/(s+t)\leq 1$ and by integrating over $z$ we get

$\begin{split}&\overline{J}(d,\beta,N,T,\mathbf{R}_{1})\\\
&=N^{2}\int_{0}^{\pi}\iint_{(s,t)\in\mathbf{R}_{1}}\frac{1}{t+s}\int_{\mathbf{B}_{2}(0)}\exp\left(-\frac{\big{|}z-D_{t}^{(k)}+D_{s}^{(\ell)}\big{|}^{2}}{2(t+s)}\right)dzdsdtd\theta\\\
&\leq CN^{2}\int_{0}^{\pi}\iint_{(s,t)\in\mathbf{R}_{1}}\frac{1}{t+s}\\\
&\hskip
28.45274pt\times\exp\left(-\frac{\left(|D^{(k)}_{t}|-|D^{(\ell)}_{s}|\right)^{2}+C|D^{(k)}_{t}|\cdot|D^{(\ell)}_{s}|\theta^{2}}{2(t+s)}+1\right)dsdtd\theta.\end{split}$

Next we factor the above exponential, incorporate $e^{1}$ into the constant,
and apply our elementary integral (5.5) to obtain

(5.11) $\begin{split}&\overline{J}(d,\beta,N,T,\mathbf{R}_{1})\\\ &\leq
CN^{2}\iint_{(s,t)\in\mathbf{R}_{1}}\frac{1}{t+s}\exp\left(-\frac{\left(|D^{(k)}_{t}|-|D^{(\ell)}_{s}|\right)^{2}}{2(t+s)}\right)\\\
&\qquad\qquad\qquad\qquad\qquad\qquad\times\left[\int_{0}^{\pi}\exp\left(-\frac{C|D^{(k)}_{t}|\cdot|D^{(\ell)}_{s}|\theta^{2}}{2(t+s)}\right)d\theta\right]dsdt\\\
&=CN^{2}\iint_{(s,t)\in\mathbf{R}_{1}}\frac{1}{\sqrt{(t+s)|D^{(k)}_{t}|\cdot|D^{(\ell)}_{s}|}}\\\
&\qquad\qquad\qquad\qquad\qquad\qquad\times\exp\left(-\frac{\left(|D^{(k)}_{t}|-|D^{(\ell)}_{s}|\right)^{2}}{2(t+s)}\right)dsdt.\end{split}$

Then we use the fact that $0\leq s\leq t$, and so
$|D_{s}^{(\ell)}|\leq|D_{t}^{(k)}|$. From (3.7) and (5.1) we have
$|D^{(k)}_{t}|=\beta^{1/4}N^{1/4}t^{3/4}$, so we get

$\begin{split}&\overline{J}(d,\beta,N,T,\mathbf{R}_{1})\\\
&\leq\tilde{C}N^{2}\iint_{(s,t)\in\mathbf{R}_{1}}t^{-\frac{1}{2}}\beta^{-\frac{1}{4}}N^{-\frac{1}{4}}t^{-\frac{3}{8}}s^{-\frac{3}{8}}\\\
&\qquad\qquad\qquad\qquad\times\exp\left(-C\frac{\beta^{\frac{1}{2}}N^{\frac{1}{2}}[t^{\frac{3}{4}}-s^{\frac{3}{4}}]^{2}}{t}\right)dsdt\\\
&\leq\tilde{C}\beta^{-\frac{1}{4}}N^{\frac{7}{4}}\int_{0}^{T}\int_{0}^{t}t^{-\frac{7}{8}}s^{-\frac{3}{8}}\\\
&\qquad\qquad\qquad\qquad\times\exp\left(-C\beta^{\frac{1}{2}}N^{\frac{1}{2}}t^{-1}[t^{\frac{3}{4}}-s^{\frac{3}{4}}]^{2}\right)dsdt.\end{split}$

Now by making a change variables to $u=s/T$ and $v=t/T$ it follows that,

(5.12) $\begin{split}&\overline{J}(d,\beta,N,T,\mathbf{R}_{1})\\\
&=\tilde{C}\beta^{-\frac{1}{4}}N^{\frac{7}{4}}\int_{0}^{1}\int_{0}^{v}(Tv)^{-\frac{7}{8}}(Tu)^{-\frac{3}{8}}\\\
&\hskip
39.83368pt\times\exp\left(-C\beta^{\frac{1}{2}}N^{\frac{1}{2}}(Tv)^{-1}[(Tv)^{\frac{3}{4}}-(Tu)^{\frac{3}{4}}]^{2}\right)d(Tu)d(Tv)\\\
&=\tilde{C}\beta^{-\frac{1}{4}}N^{\frac{7}{4}}T^{\frac{3}{4}}\int_{0}^{1}\int_{0}^{v}v^{-\frac{7}{8}}u^{-\frac{3}{8}}\exp\left(-C\beta^{\frac{1}{4}}N^{\frac{1}{4}}T^{\frac{1}{2}}v^{-1}\big{[}v^{\frac{3}{4}}-u^{\frac{3}{4}}\big{]}^{2}\right)dudv.\end{split}$

We will show that the following bound holds for all $K\geq 1$,

(5.13)
$\begin{split}\int_{0}^{1}\int_{0}^{v}v^{-\frac{7}{8}}u^{-\frac{3}{8}}\exp\left(-CKv^{-1}[v^{\frac{3}{4}}-u^{\frac{3}{4}}]^{2}\right)dudv\leq
K^{-1/2}.\end{split}$

We choose $K=\beta^{1/2}N^{1/2}T^{1/2}$ as in (5.12) so the bound in (5.13)
will give us

$\overline{J}(d,\beta,N,T,\mathbf{R}_{1})\leq C\beta^{-1}N,$

for $\beta,T\geq 1$, and this will complete the proof for $d=2$ and $d=3$.

By the mean value theorem, there exists $r\in(u,v)$ such that

$v^{\frac{3}{4}}-u^{\frac{3}{4}}=cr^{-\frac{1}{4}}(v-u)\geq
cv^{-\frac{1}{4}}(v-u).$

Hence following (5.12), it is enough to bound

(5.14)
$\begin{split}&\int_{0}^{1}\int_{0}^{v}v^{-\frac{7}{8}}u^{-\frac{3}{8}}\exp\left(-Kv^{-\frac{3}{2}}[v-u]^{2}\right)dudv\\\
&=\int_{0}^{1}\int_{0}^{v}v^{-\frac{7}{8}}(v-w)^{-\frac{3}{8}}\exp\left(-Kv^{-\frac{3}{2}}w^{2}\right)dwdv.\end{split}$

We examine the inner integral in (5.14), and write

(5.15)
$\begin{split}&\int_{0}^{v}v^{-\frac{7}{8}}(v-w)^{-\frac{3}{8}}\exp\left(-Kv^{-\frac{3}{2}}w^{2}\right)dw\\\
&=\int_{0}^{v/2}v^{-\frac{7}{8}}(v-w)^{-\frac{3}{8}}\exp\left(-Kv^{-\frac{3}{2}}w^{2}\right)dw\\\
&\quad+\int_{v/2}^{v}v^{-\frac{7}{8}}(v-w)^{-\frac{3}{8}}\exp\left(-Kv^{-\frac{3}{2}}w^{2}\right)dw\\\
&=:H_{1}(v)+H_{2}(v).\end{split}$

First dealing $H_{1}$, we have

$\begin{split}H_{1}(v)&=\int_{0}^{v/2}(v-w)^{-\frac{3}{8}}\exp\left(-Kv^{-\frac{3}{2}}w^{2}\right)dw\\\
&\leq
Cv^{-\frac{3}{8}}\int_{0}^{\infty}\exp\left(-Kv^{-\frac{3}{2}}w^{2}\right)dw\\\
&\leq Cv^{-\frac{3}{8}}K^{-1/2}v^{\frac{3}{4}}\\\ &\leq
Cv^{\frac{3}{8}}K^{-1/2},\end{split}$

using (5.5). Integrating over $v$ as well, we get

(5.16)
$\begin{split}\int_{0}^{1}v^{-\frac{7}{8}}H_{1}(v)dv&=CK^{-1/2}\int_{0}^{1}v^{-\frac{7}{8}}v^{\frac{3}{8}}dv\\\
&=CK^{-1/2}.\end{split}$

Turning to $H_{2}$, we have

$\begin{split}H_{2}(v)&=\int_{v/2}^{v}(v-w)^{-\frac{3}{8}}\exp\left(-Kv^{-\frac{3}{2}}w^{2}\right)dw\\\
&\leq\exp\left(-CKv^{-\frac{3}{2}}v^{2}\right)\int_{v/2}^{v}(v-w)^{-\frac{3}{8}}dw\\\
&\leq Cv^{\frac{5}{8}}\exp\left(-CKv^{\frac{1}{2}}\right).\end{split}$

Integrating over $v$ as well, we get

(5.17)
$\begin{split}\int_{0}^{1}v^{-\frac{7}{8}}H_{2}(v)dv&\leq\int_{0}^{1}v^{-\frac{7}{8}}v^{\frac{5}{8}}\exp\left(-CKv^{\frac{1}{2}}\right)dv\\\
&=\int_{0}^{1}v^{-\frac{1}{4}}\exp\left(-CKv^{\frac{1}{2}}\right)dv\\\
&=K^{-\frac{3}{2}}\int_{0}^{1}(K^{2}v)^{-\frac{1}{4}}\\\
&\qquad\qquad\quad\times\exp\left(-C(K^{2}v)^{\frac{1}{2}}\right)d(K^{2}v)\\\
&=K^{-\frac{3}{2}}\int_{0}^{1}r^{-\frac{1}{4}}\exp\left(-Cr^{\frac{1}{2}}\right)dr\\\
&\leq K^{-3/2},\quad\textrm{for all }K\geq 1.\end{split}$

Note that we have used

$\int_{0}^{1}x^{-\eta/2}e^{-x^{\eta}}dx<\infty\quad\textrm{for all
}\eta\in(0,2).$

From (5.14), (5.15), (5.16) and (5.17) we get (5.13) and Lemma 5.2 follows. ∎

The following technical lemma gives us some essential bounds on
$\overline{J}(d,\beta,N,T,\mathbf{R}_{2})$ which was defined in (5.10).

######  Lemma 5.3.

For $d=2,3$, there exists a constant $C_{\ref{lem-j-r2}}>0$ such that for all
$\beta\geq 1,\,T>0,\,N\in\mathbb{Z}^{+}$ the following bound holds:

$\overline{J}(d,\beta,N,T,\mathbf{R}_{2})\leq C_{\ref{lem-
j-r2}}\beta^{-1/2}N^{3/2}.$

###### Proof.

Note that on $\mathbf{R}_{2}$ since $|z|^{2}<4$ we we expect
$|z-D_{t}^{(0)}-D_{s}^{(\ell)}|^{2}$ to be of the same order of $s+t$. In this
case we get that the integral over $z$ in the right-hand side of (5.10) is
approximately 1, that is,

$\int_{\mathbf{B}_{2}(0)}\frac{1}{s+t}\exp\left(-C\frac{|z-D_{t}^{(0)}+D_{s}^{(\ell(\theta))}|^{2}}{2(s+t)}\right)dz\approx
1.$

Hence we will absorb this integral as a multiplicative factor in our bounds on
$\overline{J}(d,\beta,N,T,\mathbf{R}_{2})$ as follows,

(5.18) $\displaystyle\overline{J}(d,\beta,N,T,\mathbf{R}_{2})$
$\displaystyle=CN^{2}\int_{0}^{\pi}\iint_{\mathbf{R}_{2}}\frac{1}{t+s}$
$\displaystyle\qquad\qquad\times\int_{\mathbf{B}_{2}(0)}\exp\left(-\frac{\big{|}z-D_{t}^{(0)}+D_{s}^{(\ell)}\big{|}^{2}}{2(t+s)}\right)dzdsdtd\theta$
$\displaystyle\leq CN^{2}\int_{0}^{\pi}\iint_{\mathbf{R}_{2}}dsdtd\theta.$

Now we flesh out this heuristic discussion. From (3.7) and (5.1) we get

(5.19) $\displaystyle|D_{t}^{(0)}-D_{s}^{(\ell(\theta))}|^{2}$
$\displaystyle=\beta^{\frac{1}{2}}N^{\frac{1}{2}}\left(t^{\frac{3}{4}}-s^{\frac{3}{4}}\cos\theta\right)^{2}$
$\displaystyle\quad+\beta^{\frac{1}{2}}N^{\frac{1}{2}}\left(s^{\frac{3}{4}}\sin\theta\right)^{2}.$

Suppose we are on the region $\mathbf{R}_{2}$. Then
$|D_{t}^{(0)}-D_{s}^{(\ell(\theta))}|^{2}\leq 8$, and the terms on the right
side of (5.19), which are both positive, must each be bounded by 8. Therefore
$\beta^{1/2}N^{1/2}\left(s^{3/4}\sin\theta\right)^{2}\leq 8$ and so using
$\sin\theta\geq C\theta$ for $\theta\in[0,\pi]$ we have

(5.20) $\theta\leq Cs^{-\frac{3}{4}}\beta^{-\frac{1}{4}}N^{-\frac{1}{4}}.$

From (5.19) we also have

(5.21)
$\beta^{\frac{1}{2}}N^{\frac{1}{2}}\left(t^{\frac{3}{4}}-s^{\frac{3}{4}}\cos\theta\right)^{2}\leq
8.$

Solving (5.21) for $s$, we get

$s\geq\frac{1}{\cos\theta}\left(t^{3/4}-8^{1/2}\beta^{-1/4}N^{-1/4}\right)^{4/3}\geq\left(t^{3/4}-8^{1/2}\beta^{-1/4}N^{-1/4}\right)^{4/3}$

Therefore if $t^{3/4}>8^{1/2}\beta^{-\frac{1}{4}}N^{-\frac{1}{4}}$, or
equivalently $t>4\beta^{-\frac{1}{3}}N^{-\frac{1}{3}}$ we get

(5.22)
$0<\left(t^{\frac{3}{4}}-8^{\frac{1}{2}}N^{-\frac{1}{4}}\beta^{-\frac{1}{4}}\right)^{\frac{4}{3}}<s\leq
t.$

Together with (5.18) and (5.20) we get

(5.23) $\displaystyle\overline{J}(d,\beta,N,T,\mathbf{R}_{2})$
$\displaystyle\leq CN^{2}\int_{\theta\leq
Cs^{-\frac{3}{4}}\beta^{-\frac{1}{4}}N^{-\frac{1}{4}}}\iint_{\mathbf{R}_{2}}dsdtd\theta$
$\displaystyle\leq
C\beta^{-\frac{1}{4}}N^{\frac{7}{4}}\iint_{\mathbf{R}_{2}}s^{-\frac{3}{4}}dsdt$
$\displaystyle\leq
C\beta^{-\frac{1}{4}}N^{\frac{7}{4}}\int_{4\beta^{-\frac{1}{3}}N^{-\frac{1}{3}}}^{4}\int_{(t^{\frac{3}{4}}-8^{\frac{1}{2}}N^{-\frac{1}{4}}\beta^{-\frac{1}{4}})^{\frac{4}{3}}}^{t}s^{-\frac{3}{4}}dsdt$
$\displaystyle\quad+C\beta^{-\frac{1}{4}}N^{\frac{7}{4}}\int_{0}^{4\beta^{-\frac{1}{3}}N^{-\frac{1}{3}}}\int_{0}^{4}s^{-\frac{3}{4}}dsdt$
$\displaystyle\leq
C\beta^{-\frac{1}{4}}N^{\frac{7}{4}}\int_{4\beta^{-\frac{1}{3}}N^{-\frac{1}{3}}}^{4}\big{[}t^{\frac{1}{4}}-(t^{\frac{3}{4}}-8^{\frac{1}{2}}N^{-\frac{1}{4}}\beta^{-\frac{1}{4}})^{\frac{1}{3}}\big{]}dt$
$\displaystyle\quad+C\beta^{-\frac{1}{4}}N^{\frac{7}{4}}\int_{0}^{4\beta^{-\frac{1}{3}}N^{-\frac{1}{3}}}dt$
$\displaystyle=:H_{1}(\beta,N)+H_{2}(\beta,N).$

Now since $\beta,N\geq 1$, we have
$\beta^{-1/3}N^{-1/3}\leq\beta^{-1/4}N^{-1/4}$ and

(5.24) $H_{1}(\beta,N)\leq
C\beta^{-1/4}N^{7/4}\beta^{-1/4}N^{-1/4}=C\beta^{-1/2}N^{3/2}.$

Turning to $H_{2}(\beta,N)$, we use the fact that for every $0<x<y<\infty$,

$\begin{split}x^{1/3}-y^{1/3}&=\frac{(x^{1/3}-y^{1/3})(x^{2/3}+x^{1/3}y^{1/3}+y^{2/3})}{x^{2/3}+x^{1/3}y^{1/3}+y^{2/3}}\\\
&=\frac{x-y}{x^{2/3}+x^{1/3}y^{1/3}+y^{2/3}}\\\ &\leq\frac{x-y}{x^{2/3}}.\\\
\end{split}$

By taking

$\displaystyle x=t^{3/4}$ $\displaystyle
y=t^{3/4}-8^{1/2}N^{-1/4}\beta^{-1/4},$

and using (5.22) we get the following bound,

$\begin{split}t^{\frac{1}{4}}-(t^{\frac{3}{4}}-8^{1/2}N^{-\frac{1}{4}}\beta^{-\frac{1}{4}})^{\frac{1}{3}}&=\frac{8^{1/2}N^{-1/4}\beta^{-1/4}}{t^{1/2}}\\\
&\leq CN^{-1/4}\beta^{-1/4}t^{-1/2}.\end{split}$

Together with (5.23) it follows that,

(5.25) $\begin{split}H_{2}(\beta,N)&\leq
C\beta^{-1/4}N^{7/4}\int_{0}^{4}\beta^{-1/4}N^{-1/4}t^{-1/2}dt\\\ &\leq
C\beta^{-1/2}N^{3/2}.\end{split}$

By plugging in (5.24) and (5.25) into (5.23) we complete the proof.

This completes the proof of Lemma 5.3. ∎

The following lemma gives us some essential bounds on
$\overline{J}_{d}(\beta,N,T,\mathbf{R}_{3})$ which was defined in (5.10).

######  Lemma 5.4.

For $d=2,3$ there exists a constant $C>0$ such that the following bound hold
for $\beta\geq 1$, $T>0$ and $N\in\mathbb{Z}^{+}$,

$\overline{J}(d,\beta,N,T,\mathbf{R}_{3})\leq C_{\ref{lem-
j-r3}}\beta^{-1/2}N^{3/2}.$

###### Proof.

Recalling that $|z|<2$, we have

(5.26)
$|z-D_{t}^{(k)}+D_{s}^{(\ell)}|^{2}\geq|D_{t}^{(k)}+D_{s}^{(\ell)}|^{2}-4^{2}>\frac{1}{2}|D_{t}^{(k)}+D_{s}^{(\ell)}|^{2}$

Using (3.7), (5.1) and (5.4), for $\ell=\ell(\theta)$ we get

(5.27)
$\begin{split}|D_{t}^{(0)}+D_{s}^{(\ell)}|^{2}&\geq\big{(}|D_{t}|-|D_{s}|\big{)}^{2}+2|D_{t}|\cdot|D_{s}|\theta^{2}\\\
&\geq\beta^{1/2}N^{1/2}\left[\big{(}t^{3/4}-s^{3/4}\big{)}^{2}+2t^{3/4}s^{3/4}\theta^{2}\right].\end{split}$

From (5.26) and (5.27) and we can bound right-hand side of (5.10) as follows,

(5.28) $\begin{split}&\overline{J}(d,\beta,N,T,\mathbf{R}_{3})\\\ &\leq
CN^{2}\iint_{\mathbf{R}_{3}}\frac{1}{t+s}\exp\left(-\frac{\frac{1}{2}\beta^{\frac{1}{2}}N^{\frac{1}{2}}\left[\big{(}t^{\frac{3}{4}}-s^{\frac{3}{4}}\big{)}^{2}\right]}{2(t+s)}\right)\\\
&\quad\times\int_{0}^{\pi}\exp\left(-\frac{\beta^{\frac{1}{2}}N^{\frac{1}{2}}t^{\frac{3}{4}}s^{\frac{3}{4}}\theta^{2}}{2(t+s)}\right)d\theta
dsdt.\end{split}$

Next, we integrate inner integral on the right-hand side of (5.28) over
$\theta$ using $s+t\leq 2t$ and (5.5) to get

(5.29)
$\begin{split}\int_{0}^{\pi}\exp\left(-\frac{\beta^{\frac{1}{2}}N^{\frac{1}{2}}s^{\frac{3}{4}}t^{\frac{3}{4}}\theta^{2}}{2t}\right)d\theta&\leq
C\beta^{-\frac{1}{4}}N^{-\frac{1}{4}}s^{-\frac{3}{8}}t^{\frac{1}{8}}.\end{split}$

From (5.28) and (5.29) and by using $s+t\geq t$ we get,

(5.30) $\begin{split}&\overline{J}(d,\beta,N,T,\mathbf{R}_{3})\\\ &\leq
C\iint_{\mathbf{R}_{3}}\left(\beta^{-\frac{1}{4}}N^{-\frac{1}{4}}t^{\frac{1}{8}}s^{-\frac{3}{8}}\right)\frac{1}{t}\exp\left(-\frac{C\beta^{\frac{1}{2}}N^{\frac{1}{2}}\big{(}t^{\frac{3}{4}}-s^{\frac{3}{4}}\big{)}^{2}}{4t}\right)dsdt\\\
&\leq C\iint_{\mathbf{R}_{3}\cap\\{s\geq\varepsilon
t]\\}}\beta^{-\frac{1}{4}}N^{-\frac{1}{4}}t^{-\frac{7}{8}}s^{-\frac{3}{8}}\exp\left(-\frac{C\beta^{\frac{1}{2}}N^{\frac{1}{2}}\big{(}t^{\frac{3}{4}}-s^{\frac{3}{4}}\big{)}^{2}}{4t}\right)dsdt\\\
&\quad+C\iint_{\mathbf{R}_{3}\cap\\{s\in[0,\varepsilon
t]\\}}\beta^{-\frac{1}{4}}N^{-\frac{1}{4}}t^{-\frac{7}{8}}s^{-\frac{3}{8}}\exp\left(-\frac{C\beta^{\frac{1}{2}}N^{\frac{1}{2}}\big{(}t^{\frac{3}{4}}-s^{\frac{3}{4}}\big{)}^{2}}{4t}\right)dsdt\\\
&=:\overline{J}_{1}(d,\beta,N,T,\mathbf{R}_{3})+\overline{J}_{2}(d,\beta,N,T,\mathbf{R}_{3}),\end{split}$

where $\varepsilon\in(0,1)$.

First we consider the case where $\varepsilon t\leq s\leq t$. Using the mean
value theorem, we see that there exists $r=r(s,t)$ such that

(5.31) $t^{3/4}-s^{3/4}=Cr^{-\frac{1}{4}}(t-s)\geq Ct^{-\frac{1}{4}}(t-s).$

Using $s^{-\frac{3}{8}}\leq\varepsilon^{-\frac{3}{8}}t^{-\frac{3}{8}}$, (5.9),
(5.31) and (5.5) on (5.30) we get

(5.32) $\begin{split}&\overline{J}_{1}(d,\beta,N,T,\mathbf{R}_{3})\\\ &\leq
C(\varepsilon)\beta^{-\frac{1}{4}}N^{-\frac{1}{4}}\int_{0}^{4}t^{-\frac{5}{4}}\int_{0}^{t}\exp\left(-\frac{C\beta^{\frac{1}{2}}N^{\frac{1}{2}}\big{(}t^{\frac{3}{4}}-s^{\frac{3}{4}}\big{)}^{2}}{4t}\right)dsdt\\\
&\leq
C(\varepsilon)\beta^{-\frac{1}{4}}N^{-\frac{1}{4}}\int_{0}^{4}t^{-\frac{5}{4}}\int_{0}^{t}\exp\left(-\frac{C\beta^{\frac{1}{2}}N^{\frac{1}{2}}t^{-\frac{1}{2}}(t-s)^{2}}{4t}\right)dsdt\\\
&\leq
C(\varepsilon)\beta^{-1}N^{-\frac{1}{4}}\int_{0}^{4}t^{-\frac{5}{4}}\int_{-\infty}^{\infty}\exp\left(-\frac{1}{4}\beta^{C\frac{1}{2}}N^{\frac{1}{2}}t^{-\frac{3}{2}}(t-s)^{2}\right)dsdt\\\
&\leq
C(\varepsilon)\beta^{-\frac{1}{4}}N^{-\frac{1}{4}}\int_{0}^{4}t^{-\frac{5}{4}}\beta^{-\frac{1}{4}}N^{-\frac{1}{4}}t^{\frac{3}{4}}dt\\\
&\leq C\beta^{-\frac{1}{2}}N^{-\frac{1}{2}}\int_{0}^{1}t^{-\frac{1}{2}}dt\\\
&\leq C\beta^{-\frac{1}{2}}N^{-\frac{1}{2}}.\end{split}$

Next we consider the case of $0\leq s<\varepsilon t$. Then we have

(5.33) $\frac{\big{(}t^{\frac{3}{4}}-s^{\frac{3}{4}}\big{)}^{2}}{4t}\geq
c_{1}(\varepsilon)t^{\frac{1}{2}}.$

Using $0\leq s<\varepsilon t$ again, now together with (3.7), (5.1) and (5.2)
we get

$\begin{split}|D_{t}^{(0)}+D_{s}^{(\ell)}|^{2}&\leq\big{(}|D_{t}|-|D_{s}|\big{)}^{2}+4|D_{t}|\cdot|D_{s}|\\\
&\leq\beta^{1/2}N^{1/2}\left[\big{(}t^{3/4}-s^{3/4}\big{)}^{2}+4t^{3/4}s^{3/4}\right]\\\
&\leq(1+4\varepsilon^{3/4})\beta^{1/2}N^{1/2}t^{3/2}.\end{split}$

Recall that on $\mathbf{R}_{3}$ we have $|D_{t}^{(0)}+D_{s}^{(\ell)}|^{2}>8$
(see (5.9)), hence it follows that there exists $c_{2}(\varepsilon)>0$ such
that,

(5.34) $t\geq c_{2}(\varepsilon)\beta^{-1/3}N^{-1/3}.$

From (5.30), (5.33) and (5.34) we get

(5.35) $\begin{split}&\overline{J}_{2}(d,\beta,N,T,\mathbf{R}_{3})\\\ &\leq
C\beta^{-\frac{1}{4}}N^{-\frac{1}{4}}\int_{0}^{4}s^{-\frac{3}{8}}ds\int_{c_{2}(\varepsilon)\beta^{-\frac{1}{3}}N^{-\frac{1}{3}}}^{4}t^{-\frac{7}{8}}\exp\left(-C(\varepsilon)\beta^{\frac{1}{2}}N^{\frac{1}{2}}t^{\frac{1}{2}}\right)dt\\\
&\leq
C\beta^{-\frac{1}{4}}N^{-\frac{1}{4}}\exp\left(-C(\varepsilon)\beta^{\frac{1}{6}}N^{\frac{1}{6}}\right)\int_{0}^{4}t^{-\frac{7}{8}}dt\\\
&\leq C\beta^{-\frac{1}{2}}N^{-\frac{1}{2}}.\end{split}$

Plugging in (5.32) and (5.35) into (5.30) and multiplying by $N^{2}$, we get
$\beta^{-1/2}N^{3/2}$ as required. This finishes the proof of Lemma 5.4.

∎

###### Proof of Proposition 4.1 for $J_{2}(d,\beta,N,T)$.

The bounds on $J_{2}(d,\beta,N,T)$ for $d=2,3$ follow directly from Lemmas
5.2–5.4. ∎

##  6\. Self-intersection occupation measure for $d=2,3$

In this section we derive the bounds on $J_{d,1}(\beta,N,T)$ from Proposition
4.1 for $d=2,3$. Recall that in this case we are dealing with the occupation
measure terms related to a single branch which is intersects with itself,
therefore there are $N$ such contributions.

We recall that $f^{(k,k,\alpha)}_{t,s}$ is the probability density function of
$B^{(k)}_{t}-B^{(k)}_{s}$ under $P^{\lambda}_{T}$. Instead of working with
$P^{\lambda}_{T}$, we will work with $P_{T}$ and consider processes
$B^{(k)}_{t}+D^{(k)}_{t}$, where $D^{(k)},D^{(\ell)}$ are the respective drift
processes. First note that

$B^{(k)}_{t}-B^{(k)}_{s}\sim\mathcal{N}(0,t-s),$

where as before $\mathcal{N}(\cdot,\cdot)$ stands for the $d$-dimensional
normal distribution.

Recall that the drift magnitudes are given by (5.1). Without loss of
generality, we can assume that $s<t$ and that $D^{(k)}$ points in the
$\mathbf{e}_{1}$ direction, where $\mathbf{e}_{1}$ is the first coordinate
vector.

For simplicity we define

(6.1) $\zeta=\beta^{\frac{1}{4}}N^{\frac{1}{4}},$

then using (5.1) and (3.7) we get

(6.2)
$\begin{split}f^{(k,k,\alpha)}_{t,s}(z)&=\frac{1}{\left(2\pi(t-s)\right)^{-d/2}}\exp\left(-\frac{|z-(\zeta
t^{\frac{3}{4}}-\zeta s^{\frac{3}{4}})\mathbf{e}_{1}|^{2}}{2(t-s)}\right)\\\
&=\left(2\pi(t-s)\right)^{-d/2}\exp\left(-\frac{(z_{1}-\zeta
t^{\frac{3}{4}}+\zeta s^{\frac{3}{4}})^{2}}{2(t-s)}\right)\\\ &\hskip
113.81102pt\times\prod_{k=2}^{d}\exp\left(-\frac{z_{k}^{2}}{2(t-s)}\right).\end{split}$

For convenience we again split the area of integration in $J_{1}(\beta,N,T)$,
$\\{(s,t):\,0\leq s\leq t\leq T\\}$ to the following subregions.

(6.3) $\displaystyle\hat{\textbf{R}}_{1}$
$\displaystyle=\\{(s,t)\in[0,T]^{2}\,:t^{\frac{3}{4}}-s^{\frac{3}{4}}>C_{(\ref{r-i-j1})}\zeta^{-1}\\},$
$\displaystyle\hat{\mathbf{R}}_{2}$
$\displaystyle=\\{(t,s)\in[0,T]^{2}:t^{\frac{3}{4}}-s^{\frac{3}{4}}<C_{(\ref{r-i-j1})}\zeta^{-1}\\},$

where $C_{(\ref{r-i-j1})}>0$ is a constant to be specified later.

Define

(6.4)
$\hat{J}(d,\beta,N,T,\hat{\mathbf{R}}_{i})=N\iint_{\hat{\textbf{R}}_{i}}\int_{\mathbf{B}_{2}(0)}f^{(1,1,\alpha)}_{t,s}(z)dzdsdt.$

Since the self-occupation measure is similar for all branches of the polymers
it follows from (4.2),

(6.5) $J_{1}(d,\beta,N,T)\leq
C\sum_{i=1}^{2}\hat{J}(d,\beta,N,T,\hat{\mathbf{R}}_{i}).$

We introduce a few technical lemmas that will help us to bound
$J_{1}(d,\beta,N,T)$.

######  Lemma 6.1.

For $d=2,3$ there exist positive constants $C_{(\ref{r-i-j1})}$ and
$C_{\ref{lem-j1-1}}$ such that the following bound holds for $1\leq T\leq
N,\,\beta\geq 1,\,N\in\mathbb{Z}^{+}$:

$\hat{J}(d,\beta,N,T,\hat{\mathbf{R}}_{1})\leq
C_{\ref{lem-j1-1}}\beta^{-\frac{1}{2}}N^{\frac{3}{2}}T^{\frac{1}{2}}\log(\beta
T).$

###### Proof.

From (6.3) we can choose $C_{(\ref{r-i-j1})}$ in the definition of
$\hat{\mathbf{R}}_{1}$ such that

(6.6) $(z_{1}-\zeta t^{\frac{3}{4}}+\zeta
s^{\frac{3}{4}})^{2}\geq\frac{1}{2}(\zeta t^{3/4}-\zeta
s^{3/4})^{2},\quad\textrm{for all }t,s\in\hat{\mathbf{R}}_{1}.$

From (6.2), (6.4) and (6.6) it follows that

(6.7) $\begin{split}\hat{J}(d,\beta,N,T,\hat{\mathbf{R}}_{1})&\leq
N\iint_{\hat{\mathbf{R}}_{1}}\frac{1}{\sqrt{2\pi(t-s)}}\int_{|z_{1}|<2}e^{-\frac{\big{(}z_{1}-\zeta
t^{\frac{3}{d+2}}+\zeta s^{\frac{3}{4}}\big{)}^{2}}{2(t-s)}}dz_{1}\\\ &\hskip
56.9055pt\times\prod_{k=2}^{d}\left(\int_{|z_{k}|<2}\frac{1}{\sqrt{2\pi(t-s)}}e^{-\frac{z_{k}^{2}}{2(t-s)}}dz_{k}\right)dsdt\\\
&\leq
N\iint_{\hat{\mathbf{R}}_{1}}\frac{1}{\sqrt{2\pi(t-s)}}\int_{|z_{1}|<2}e^{-\frac{\big{(}\zeta
t^{\frac{3}{4}}-\zeta s^{\frac{3}{4}}\big{)}^{2}}{4(t-s)}}dz_{1}\\\ &\hskip
56.9055pt\times\prod_{k=2}^{d}\left(\int_{z_{k}\in\mathbf{R}}\frac{1}{\sqrt{2\pi(t-s)}}e^{-\frac{z_{k}^{2}}{2(t-s)}}dz_{k}\right)dsdt\\\
&\leq
N\iint_{\hat{\mathbf{R}}_{1}}\frac{C}{\sqrt{2\pi(t-s)}}\exp\left(-\frac{\zeta^{2}\big{(}t^{\frac{3}{4}}-s^{\frac{3}{4}}\big{)}^{2}}{4(t-s)}\right)dsdt.\end{split}$

Since $0<s<t$, the mean value theorem implies that for some $r\in(s,t)$ we
have

$\begin{split}\frac{\big{(}t^{\frac{3}{4}}-s^{\frac{3}{4}}\big{)}^{2}}{t-s}&=\left(\frac{t^{\frac{3}{4}}-s^{\frac{3}{4}}}{t-s}\right)^{2}(t-s)=\big{(}r^{-\frac{1}{4}}\big{)}^{2}(t-s)\\\
&\geq t^{-\frac{1}{2}}(t-s),\end{split}$

where in the last line follows because $r<t$.

We therefore have

$\begin{split}\hat{J}(d,\beta,N,T,\hat{\mathbf{R}}_{1})\leq
N\iint_{\hat{\mathbf{R}}_{1}}\frac{C}{\sqrt{2\pi(t-s)}}\exp\left(-\frac{1}{4}t^{-\frac{1}{2}}(t-s)\right)dsdt.\end{split}$

Using the fact that $0\leq s<t\leq T$ and

$\int_{\mathbf{B}_{2}(0)}f^{(k,k,\alpha)}_{t,s}(z)dz\leq 1,$

together with (6.4) we arrive at

(6.8) $\displaystyle\hat{J}(d,\beta,N,T,\hat{\mathbf{R}}_{1})$
$\displaystyle\leq
N\iint_{\hat{\mathbf{R}}_{1}}\left(\left(\frac{C}{\sqrt{2\pi(t-s)}}\exp\left(-\frac{1}{4}T^{-\frac{1}{2}}\zeta^{2}(t-s)\right)\right)\wedge
1\right)dsdt.$

Define

(6.9) $\displaystyle\gamma_{1}=\frac{5}{4},\qquad\gamma_{2}=\frac{3}{4},$

and

(6.10)
$\gamma:=\frac{1}{4}(N\beta)^{-\frac{1}{2}}T^{\frac{1}{2}}\log(T^{\gamma_{1}}\beta^{\gamma_{2}}).$

From (6.1) and (6.10) we get for all $t-s>\gamma$,

(6.11)
$\begin{split}&\frac{1}{\sqrt{2\pi(t-s)}}\exp\left(-\frac{1}{4}T^{-\frac{1}{2}}\zeta^{2}(t-s)\right)\\\
&\leq
C\gamma^{-1/2}\exp\left(-\frac{1}{4}T^{-\frac{2d-2}{d+2}}\zeta^{2}\gamma\right)\\\
&\leq
C\gamma^{-1/2}\exp\left(-\frac{1}{4}T^{-\frac{1}{2}}N^{\frac{1}{2}}\beta^{\frac{1}{2}}\gamma\right)\\\
&\leq
C(N\beta)^{\frac{1}{4}}T^{-\frac{1}{4}}\log(T^{\gamma_{1}}\beta^{\gamma_{2}})^{-1/2}(T^{\gamma_{1}}\beta^{\gamma_{2}})^{-1}\\\
&\leq C\beta^{-\frac{1}{2}}N^{\frac{1}{2}}T^{-\frac{3}{2}}(\log(\beta
T))^{-1/2},\end{split}$

where we have plugged in the values of $\gamma_{i}$ from (6.9) in the last
inequality.

From (6.8), (6.11) and since $0<s<t<T$ it follows that

$\begin{split}&\hat{J}(d,\beta,N,T,\hat{\mathbf{R}}_{1})\\\ &\leq
CN\int_{0}^{T}\int_{0}^{t-\gamma}\frac{1}{\sqrt{2\pi(t-s)}}\exp\left(-\frac{1}{4}t^{-\frac{1}{2}}(t-s)\right)dsdt\\\
&\quad+CN\int_{0}^{T}\int_{t-\gamma}^{t}1dsdt\\\ &\leq
CN\int_{0}^{T}\int_{0}^{T}\beta^{-\frac{1}{2}}N^{\frac{1}{2}}T^{-\frac{3}{2}}(\log(\beta
T))^{-1/2}dsdt+CNT\gamma\\\ &\leq
C\beta^{-\frac{1}{2}}N^{\frac{3}{2}}T^{\frac{1}{2}}(\log(\beta
T))^{-1/2}+CNT(N\beta)^{-\frac{1}{2}}T^{\frac{1}{2}}\log(T\beta)\\\ &\leq
C\beta^{-\frac{1}{2}}N^{\frac{3}{2}}T^{\frac{1}{2}}(\log(\beta
T))^{-1/2}+C\beta^{-\frac{1}{2}}N^{\frac{1}{2}}T^{\frac{3}{2}}\log(T\beta).\end{split}$

By choosing

(6.12) $T\leq N,$

we get

$\begin{split}\hat{J}(d,\beta,N,T,\hat{\mathbf{R}}_{1})\leq
C\beta^{-\frac{1}{2}}N^{\frac{3}{2}}T^{\frac{1}{2}}\log(\beta
T),\quad\textrm{for all }T,\beta\geq 1,\end{split}$

This completes the proof of Lemma 6.1. ∎

######  Lemma 6.2.

For $d=2,3$ there exists a constant $C_{\ref{lem-j1-2}}>0$ such that the
following bound holds for $\beta\geq 1$, $T>0$ and $N\in\mathbb{Z}^{+}$:

$\hat{J}(d,\beta,N,T,\hat{\mathbf{R}}_{2})\leq
C_{\ref{lem-j1-2}}\beta^{-2/3}N^{1/3}.$

###### Proof.

From (6.1) and (6.3) it follows that on $\hat{\mathbf{R}}_{2}$ we have

(6.13) $\displaystyle s^{\frac{3}{d+2}}$ $\displaystyle\leq
t^{\frac{3}{d+2}}\leq
C\beta^{-\frac{1}{d+2}}N^{-\frac{1}{d+2}}+s^{\frac{3}{d+2}},$ $\displaystyle
s$ $\displaystyle\leq C\beta^{-\frac{1}{3}}N^{-\frac{1}{3}}.$

By integrating over $z$ in the right-hand side of (6.4) and then using (6.13)
we get

(6.14) $\displaystyle\hat{J}(d,\beta,N,T,\hat{\mathbf{R}}_{2})$
$\displaystyle\leq N|\hat{\mathbf{R}}_{2}|$ $\displaystyle\leq
N\int_{0}^{C\beta^{-\frac{1}{3}}N^{-\frac{1}{3}}}\int_{0}^{(C\beta^{-\frac{1}{4}}N^{-\frac{1}{4}}+s^{\frac{3}{4}})^{\frac{4}{3}}}dtds$
$\displaystyle\leq
N\int_{0}^{C\beta^{-\frac{1}{3}}N^{-\frac{1}{3}}}(C\beta^{-\frac{1}{4}}N^{-\frac{1}{4}}+s^{\frac{3}{4}})^{\frac{4}{3}}ds$
$\displaystyle\leq
CN\int_{0}^{C\beta^{-\frac{1}{3}}N^{-\frac{1}{3}}}(\beta^{-\frac{1}{4}}N^{-\frac{1}{4}})^{\frac{4}{3}}ds$
$\displaystyle\leq C\beta^{-\frac{2}{3}}N^{\frac{1}{3}}.$

This completes the proof of Lemma 6.2. ∎

###### Proof of Proposition 4.1 for $J_{1}(d,\beta,N,T)$.

The bounds on $J_{1}(d,\beta,N,T)$ for $d=2,3$ follow directly from Lemmas 6.1
and 6.2 and from (6.5). ∎

##  7\. Proof of Proposition 3.2 for $d=1$

Following (3.7) we choose

(7.1) $\kappa=\beta^{1/3}N^{1/3},\quad\alpha=0,$

for the parameters of drift magnitude given by (5.1). Note that in the one-
dimensional case we give all $N$ particles drift in the positive direction of
$\mathbb{R}$.

###  7.1. Cross-intersection occupation measure

We follow the argument in Section 5, with the change that under
$\mathbb{P}_{T}^{\lambda}$ all drifts go in the positive direction on
$\mathbb{R}$. Recall that there are $O(N^{2})$ pairs of particles. We
therefore have as in (4.2) as follows,

(7.2) $\displaystyle
J_{2}(1,\beta,N,T):=CN^{2}\int_{0}^{T}\int_{0}^{T}\int_{\mathbf{B}_{2}(0)}f^{(k,\ell,\alpha)}_{t,s}(z)dzdsdt$
$\displaystyle\leq
CN^{2}\int_{0}^{T}\int_{0}^{T}\left(\left(\frac{1}{(t+s)^{1/2}}\int_{|z_{1}|<2}e^{-\frac{(z_{1}-\beta^{1/3}N^{1/3}(t-s))^{2}}{2(t+s)}}dz_{1}\right)\wedge
1\right)dsdt.$

The following lemma give us the essential bound on $J_{2}(1,\beta,N,T)$.

######  Lemma 7.1.

There exists a constant $C_{\ref{dim1-lem-j2}}>0$ such that for all
$\beta,T\geq 1$ and $N\in\mathbb{Z}^{+}$ the following bound holds:

$J_{2}(1,\beta,N,T)\leq C_{\ref{dim1-lem-j2}}\beta^{2/3}N^{5/3}T.$

###### Proof.

From we have (7.2),

(7.3) $\displaystyle J_{2}(1,\beta,N,T)$ $\displaystyle\leq
CN^{2}\int_{0}^{T}\int_{0}^{T}\left(\left(\frac{1}{(t+s)^{1/2}}\int_{|z_{1}|<2}e^{-\frac{(z_{1}-\beta^{1/3}N^{1/3}(t-s))^{2}}{2(t+s)}}dz_{1}\right)\wedge
1\right)dsdt.$

We fix $C_{0}>2$ satisfying,

(7.4)
$(2-\beta^{1/3}N^{1/3}(t-s))^{2}\geq\frac{1}{2}\beta^{2/3}N^{2/3}(t-s)^{2},\quad\textrm{for
all }t-s>C_{0}\beta^{-\frac{1}{3}}N^{-\frac{1}{3}},$

and consider the following regions:

(7.5) $\displaystyle\hat{\mathbf{R}}_{1}$
$\displaystyle=\\{(t,s)\in[0,T]^{2}:\beta^{1/3}N^{1/3}(t-s)>C_{0}\\}$
$\displaystyle\hat{\mathbf{R}}_{2}$
$\displaystyle=\\{(t,s)\in[0,T]^{2}:\beta^{1/3}N^{1/3}(t-s)\leq C_{0}\\}.$

We define

(7.6)
$\tilde{J}_{2}(1,\beta,N,T,\hat{\mathbf{R}}_{i}):=N^{2}\iint_{\hat{\mathbf{R}}_{i}}\int_{\mathbf{B}_{2}(0)}f^{(1,1,\alpha)}_{t,s}(z)dzdsdt,\quad
i=1,2,$

and note that

(7.7)
$J_{2}(1,\beta,N,T)\leq\sum_{i=1}^{2}\tilde{J}_{2}(1,\beta,N,T,\hat{\mathbf{R}}_{i}).$

From (7.3), (7.4), (7.5) and (7.6) we get that

(7.8) $\displaystyle\tilde{J}_{2}(1,\beta,N,T,\hat{\mathbf{R}}_{1})$
$\displaystyle\leq
CN^{2}\int_{0}^{T}\int_{0}^{t}\left(\frac{1}{(2\pi(t+s))^{1/2}}e^{-\beta^{2/3}N^{2/3}\frac{(t-s)^{2}}{4(t+s)}}\right)\wedge
1dsdt$ $\displaystyle\leq
CN^{2}\int\int_{t-s>\delta}\left(\frac{1}{(2\pi(t+s))^{1/2}}e^{-\beta^{2/3}N^{2/3}\frac{(t-s)^{2}}{4(t+s)}}\right)\wedge
1dsdt$
$\displaystyle\quad+CN^{2}\int\int_{0<t-s\leq\delta}\left(\frac{1}{(2\pi(t+s))^{1/2}}e^{-\beta^{2/3}N^{2/3}\frac{(t-s)^{2}}{4(t+s)}}\right)\wedge
1dsdt$ $\displaystyle\leq
CN^{2}\int\int_{t-s>\delta}\frac{1}{(2\pi(t+s))^{1/2}}e^{-\beta^{2/3}N^{2/3}\frac{(t-s)^{2}}{4(t+s)}}dsdt+CN\int_{0}^{T}\int_{t-\delta}^{t}dsdt$
$\displaystyle=:H_{1}(\beta,N,T)+H_{2}(\beta,N,T),$

where we define

(7.9) $\delta=\beta^{-1/3}N^{-1/3}.$

First we note that

(7.10) $H_{2}(\beta,N,T)\leq CN^{2}T\delta\leq\beta^{-1/3}N^{5/3}T.$

Next we deal with (A). Let us change variables to

$\displaystyle a=t-s,$ $\displaystyle b=t+s$

and absorb the Jacobian determinant into the constant $C$. We find

(7.11) $\begin{split}H_{1}(\beta,N,T)&\leq CN^{2}\int_{0}^{2T}\frac{1}{(2\pi
b)^{1/2}}\left(\int_{\delta}^{\infty}e^{-\beta^{2/3}N^{2/3}\frac{a^{2}}{4b}}da\right)db.\end{split}$

Dealing with the inner integral, we use the bound on the integral over the
Gaussian density to get,

(7.12) $\begin{split}\frac{1}{(2\pi
b)^{1/2}}\int_{\delta}^{\infty}e^{-\beta^{2/3}N^{2/3}\frac{a^{2}}{4b}}da&\leq\frac{N^{-1/3}\beta^{-1/3}}{(2\pi
bN^{-2/3}\beta^{-2/3})^{1/2}}\int_{-\infty}^{\infty}e^{-\frac{a^{2}}{4bN^{-2/3}\beta^{-2/3}}}da\\\
&\leq CN^{-1/3}\beta^{-1/3}.\end{split}$

Applying (7.12) to the inner integral in (7.11) with we get

(7.13) $\begin{split}H_{1}(\beta,N,T)&\leq
CN^{2}\int_{0}^{2T}CN^{-1/3}\beta^{-1/3}db\\\ &\leq
C\beta^{-1/3}N^{5/3}T.\end{split}$

Plugging-in our estimates from (7.10) and (7.13) to (7.8) we get

(7.14) $\tilde{J}_{2}(1,\beta,N,T,\hat{\mathbf{R}}_{1})\leq
C\beta^{-1/3}N^{5/3}T.$

From (7.5) and (7.6) it follows that

(7.15) $\displaystyle\tilde{J}_{2}(\beta,N,T,\hat{\mathbf{R}}_{2})$
$\displaystyle\leq N^{2}|\hat{\mathbf{R}}_{2}|$ $\displaystyle\leq
CN^{2}\beta^{-1/3}N^{-1/3}T$ $\displaystyle\leq C\beta^{-1/3}N^{5/3}T.$

From (7.7), (7.14) and (7.15) we get Lemma 7.1. ∎

###  7.2. Self-intersection occupation measure

Using (5.1) and (7.1) we deduce by following similar lines as in (6.2), (6.3)
and (6.5), we need to bound the following integral:

(7.16)
$\begin{split}J_{1}(1,\beta,N,T)&:=N\int_{0}^{T}\int_{0}^{T}\int_{\mathbf{B}_{2}(0)}f^{(1,1,\alpha)}_{t,s}(z)dzdsdt\\\
&=N\int_{0}^{T}\int_{0}^{T}\frac{1}{(2\pi(t-s))^{1/2}}\int_{|z|<2}e^{-\frac{(z-\beta^{1/3}N^{1/3}(t-s))^{2}}{2(t-s)}}dzdsdt.\end{split}$

In the following lemma we derive the bound on $J_{1}(\beta,N,T)$.

######  Lemma 7.2.

There exists a constant $C_{\ref{dim1-lem-j1}}>0$ such that for all
$\beta,T\geq 1$ and $N\in\mathbb{Z}^{+}$ the following bound holds:

$J_{1}(1,\beta,N,T)\leq C_{\ref{dim1-lem-j1}}\beta^{2/3}N^{2/3}T.$

###### Proof.

Recalling $\hat{\mathbf{R}}_{i}$ defined in (7.5) we denote

(7.17)
$\tilde{J}_{1}(1,\beta,N,T,\hat{\mathbf{R}}_{i}):=N\iint_{\hat{\mathbf{R}}_{i}}\int_{\mathbf{B}_{2}(0)}f^{(1,1,\alpha)}_{t,s}(z)dzdsdt,\quad
i=1,2,$

and note that

$\tilde{J}_{1}(1,\beta,N,T)\leq\sum_{i=1}^{2}\tilde{J}_{1}(1,\beta,N,T,\hat{\mathbf{R}}_{i}).$

Using the fact that $\int_{\mathbf{B}_{2}(0)}f^{(1,1,\alpha)}_{t,s}(z)dz\leq
1$ and (7.4), we get for $\delta$ as in (7.9),

(7.18) $\begin{split}\tilde{J}_{1}(1,\beta,N,T,\mathbf{R_{1}})&\leq
CN\int\int_{\hat{\mathbf{R}}_{1}\cap\\{t-s>\delta\\}}\frac{1}{2\pi(t-s)}e^{-\frac{\beta^{2/3}N^{2/3}(t-s)^{2}}{4(t-s)}}dsdt\\\
&\quad+CN\int\int_{\hat{\mathbf{R}}_{1}\cap\\{t-s\leq\delta\\}}dsdt.\end{split}$

Note that this integral is similar to (7.8) (with a difference by a factor of
$N$) hence we get the bound

(7.19) $\tilde{J}_{1}(1,\beta,N,T,\hat{\mathbf{R}}_{1})\leq
C\beta^{2/3}N^{2/3}T.$

Using again the trivial bound on the integral over $f^{(1,1,\alpha)}$ we get
from (7.17) and (7.5) that

(7.20) $\begin{split}\tilde{J}_{1}(1,\beta,N,T,\hat{\mathbf{R}}_{2})&\leq
C\iint_{\hat{\mathbf{R}}_{2}}dsdt\leq C\beta^{-1/3}N^{2/3}T,\end{split}$

where we used a similar bound as in (7.15) with a difference of a factor of
$N$ in the last inequality.

From (7.19), (7.20) and (7.17) we get the result of Lemma 7.2. ∎

Now we are ready to prove Theorem 2.2 for $d=1$.

###### Proof of Proposition 3.2 for $d=1$.

The proof follows similar lines to the proof in the case where $d=2,3$ only we
now use Lemmas 7.1 and 7.2 instead of Proposition 4.1. ∎

## References

  * [1] N. H. Aloorkar, A. S. Kulkarmi, R. A. Patil, and D. J. Ingale.  Star polymers: an overview.  Int. J. Pharm. Sci. Nanotech. 5:1675–1684, 2012.
  * [2] R. Bauerschmidt, D. C. Brydges, and G. Slade.  Introduction to a renormalisation group method, volume 2242 of Lecture Notes in Mathematics.  Springer, Singapore, 2019.
  * [3] R. Bauerschmidt, H. Duminil-Copin, J. Goodman, and G. Slade.  Lectures on self-avoiding walks.  In Probability and statistical physics in two and more dimensions, volume 15 of Clay Math. Proc., pages 395–467. Amer. Math. Soc., Providence, RI, 2012.
  * [4] M. Bishop, M. H. Kalos, A. D. Sokal, and H. L. Frisch.  Scaling in multichain polymer systems in two and three dimensions.  The Journal of Chemical Physics, 79(7):3496–3499, 1983.
  * [5] E. Bolthausen.  On self-repellent one-dimensional random walks.  Probab. Theory Related Fields, 86(4):423–441, 1990.
  * [6] M. Daoud and J. P. Cotton.  Star shaped polymers : a model for the conformation and its concentration dependence.  Journal de Physique, 43(3):531–538, 1982.
  * [7] F. den Hollander.  Random polymers, volume 1974 of Lecture Notes in Mathematics.  Springer-Verlag, Berlin, 2009.  Lectures from the 37th Probability Summer School held in Saint-Flour, 2007.
  * [8] M. Doi and S. F. Edwards.  The Theory of Polymer Dynamics.  Oxford University Press, Walton Street, Oxford, 1986.
  * [9] H. Duminil-Copin and S. Smirnov.  The connective constant of the honeycomb lattice equals $\sqrt{2+\sqrt{2}}$.  Ann. of Math. (2), 175(3):1653–1665, 2012.
  * [10] M. Fixman.  Radius of Gyration of Polymer Chains.  The Journal of Chemical Physics, 36(2):306-–310, 1962.
  * [11] G. Giacomin.  Random polymer models.  Imperial College Press, London, 2007.
  * [12] A. Greven and F. den Hollander.  A variational characterization of the speed of a one-dimensional self-repellent random walk.  Ann. Appl. Probab., 3(4):1067–1099, 1993.
  * [13] T. Hara and G. Slade.  The lace expansion for self-avoiding walk in five or more dimensions.  Rev. Math. Phys., 4(2):235–327, 1992.
  * [14] G. Lawler, O. Schramm and W. Werner.  On the scaling limit of planar self-avoiding walk.  Fractal Geometry and applications: a jubilee of Benôit Mandelbrot, Part 2, 339–364.  Proc. Sympos. Pure. Math. 72, Part 2, Amer. Math. Soc. Providence, RI 2004.
  * [15] N. Madras and G. Slade.  The self-avoiding walk.  Modern Birkhäuser Classics. Birkhäuser/Springer, New York, 2013\.  Reprint of the 1993 original.
  * [16] G. Slade and R. van der Hofstad.  The lace expansion on a tree with application to networks of self-avoiding walks.  Adv. in Appl. Math., 30(3):471–528, 2003.
  * [17] P. A. Tikhonov, N. G. Vasilenko, and A. M. Muzafarov.  Multiarm Star Polymers. Fundamental Aspects. A Review.  Dokl. Chem. 496:1–17, 2021.
  * [18] R. van der Hofstad, F. den Hollander, and W. König.  Central limit theorem for the Edwards model.  Ann. Probab., 25(2):573–597, 1997.
  * [19] Remco van der Hofstad, Frank den Hollander, and Wolfgang König.  Large deviations for the one-dimensional Edwards model.  Ann. Probab., 31(4):2003–2039, 2003.
  * [20] R. van der Hofstad and W. König.  A survey of one-dimensional random polymers.  J. Statist. Phys., 103(5-6):915–944, 2001.