Then, $\nu^{Y}<\infty$ almost surely. Further, there exist positive constants
$C_{4},C_{5}$ such that for any $y>0$ and any integer $n$,

${\mathbb{P}}(\nu^{Y}>n\mid Y_{0}=y)\leq\frac{C_{4}+C_{5}y}{\sqrt{n}}~{}.$

By Proposition 5.1, the four hypotheses $(i)$-$(iv)$ of Corollary 6.2 are
satisfied by the process $\\{Z_{\ell}:\ell\geq 0\\}$ defined in (47) and this
gives us the required decay estimate in terms of number of (joint) renewal
steps before coalescing. Using this estimate, we proceed to establish Theorem
6.1.

###### Proof of Theorem 6.1.

It is easy to observe that
$h^{\sigma_{\ell}}({\mathbf{x}}_{1})=h^{\sigma_{\ell}}({\mathbf{x}}_{2})$
implies that $h^{m}({\mathbf{x}}_{1})=h^{m}({\mathbf{x}}_{2})$ for some $m$
such that $m\leq\sigma_{\ell}$. In other words

$T_{\nu}:=\min\\{\sigma_{\ell}:h^{\sigma_{\ell}}({\mathbf{x}}_{1})=h^{\sigma_{\ell}}({\mathbf{x}}_{2})\\}$

dominates the actual coalescing time $T({\mathbf{x}}_{1},{\mathbf{x}}_{2})$ of
the two paths. Consider an i.i.d. sequence $\\{W_{\ell}:\ell\geq 1\\}$ with
exponentially decaying tail such that for each $\ell\geq 1$ the conditional
distribution of
$(\sigma_{\ell+1}({\mathbf{x}}_{1},{\mathbf{x}}_{2})-\sigma_{\ell}({\mathbf{x}}_{1},{\mathbf{x}}_{2}))\mid{\cal
S}_{\ell}$ is stochastically dominated by $W_{\ell+1}$. Since the family
$\\{(\sigma_{\ell+1}({\mathbf{x}}_{1},{\mathbf{x}}_{2})-\sigma_{\ell}({\mathbf{x}}_{1},{\mathbf{x}}_{2})):\ell\geq
1\\}$ has strong uniform exponential tail decay (see Remark 5.3), we can
always construct such an i.i.d. sequence $\\{W_{\ell}:\ell\geq 1\\}$. Choose
$c=1/{\mathbb{E}}(2W_{1})$ and we have,

$\displaystyle{\mathbb{P}}(T_{\nu}>t)$
$\displaystyle\leq{\mathbb{P}}\Big{(}\sum_{\ell=1}^{\lfloor
ct\rfloor+1}W_{\ell}\geq t\Big{)}+{\mathbb{P}}(\nu>ct)$
$\displaystyle\leq{\mathbb{P}}\Big{(}\sum_{\ell=1}^{\lfloor
ct\rfloor+1}(W_{\ell}-{\mathbb{E}}(W_{1}))\geq
t(1-c{\mathbb{E}}(W_{1}))\Big{)}+\frac{C_{0}}{\sqrt{ct}}({\mathbf{x}}_{2}(1)-{\mathbf{x}}_{1}(1))$
$\displaystyle\leq\frac{\text{Var}\Big{(}\sum_{\ell=1}^{\lfloor
ct\rfloor+1}W_{\ell}\Big{)}}{(t(1-c{\mathbb{E}}(W_{1}))^{2}}+\frac{C_{0}}{\sqrt{ct}}({\mathbf{x}}_{2}(1)-{\mathbf{x}}_{1}(1))$
$\displaystyle\leq\frac{(\lfloor
ct\rfloor+1)\text{Var}(W_{1})}{(t/2)^{2}}+\frac{C_{0}}{\sqrt{ct}}({\mathbf{x}}_{2}(1)-{\mathbf{x}}_{1}(1))$
$\displaystyle\leq\frac{C_{1}}{\sqrt{t}}({\mathbf{x}}_{2}(1)-{\mathbf{x}}_{1}(1)),$

for a suitable choice of constant $C_{1}>0$. This completes the proof. ∎

We observe that Theorem 6.1 is applicable for PH paths starting from
${\mathbf{x}}_{1},{\mathbf{x}}_{2}\in{\mathbb{Z}}^{2}$ with
${\mathbf{x}}_{1}(2)={\mathbf{x}}_{2}(2)$ and with no information about the
collection of random vectors
$\\{\Gamma_{\mathbf{w}}:{\mathbf{w}}\in\mathbb{H}^{-}({\mathbf{x}}_{1}(2))\\}$.
On the other hand, while applying this result we need to estimate coalescing
time of two PH paths after they come close enough. By that time we gather
enough information about a large set of explored random vectors. Therefore, we
need to develop a modified version of Theorem 6.1 which would be applicable in
such situations. Before ending this section, we state such a version of
Theorem 6.1 that would be more useful while proving finite dimensional
convergence to coalescing Brownian motions. In order to do that we need to
introduce some notations.

Fix any $\beta\in(0,1/2)$ and lattice points
${\mathbf{w}}_{1},{\mathbf{w}}_{2}\in{\mathbb{Z}}^{2}$ with
${\mathbf{w}}_{1}(2)={\mathbf{w}}_{2}(2)=0$ and
${\mathbf{w}}_{1}(1)<{\mathbf{w}}_{2}(1)$. Let ${\cal
R}\subset\mathbb{H}^{-}(0)$ be a bounded region. Proposition 6.3 gives us that
given any realisations of the collection of random vectors
$\\{\Gamma_{\mathbf{w}}:{\mathbf{w}}\in{\cal R}_{n}\\}$ satisfying that
$\tilde{{\mathbf{w}}}\in\mathbb{H}^{-}(n^{\beta})$ for all
${\mathbf{w}}\in{\cal R}$, the same tail estimate for coalescing time of PH
paths as in Theorem 6.1 holds. Proposition 6.3 would be more useful for
proving convergence to coalescing Brownian motions.

######  Proposition 6.3.

For ${\mathbf{w}}_{1},{\mathbf{w}}_{2}$ chosen as above, there exists
$C_{0}>0$ (not depending on ${\mathbf{w}}_{1},{\mathbf{w}}_{2}$) such that for
all large $n$ we have

$\displaystyle{\mathbb{P}}\Bigl{(}T({\mathbf{w}}_{1},{\mathbf{w}}_{2})>n\mid\bigl{(}\\{\Gamma_{\mathbf{w}}:{\mathbf{w}}\in{\cal
R}\\}$ $\displaystyle\text{ such that
}\tilde{{\mathbf{w}}}\in\mathbb{H}^{-}(n^{\beta})\text{ for all
}{\mathbf{w}}\in{\cal R}\bigr{)}\Bigr{)}$
$\displaystyle\leq\frac{C_{0}({\mathbf{w}}_{2}(1)-{\mathbf{w}}_{1}(1))}{\sqrt{n}}.$

###### Proof.

Choose $\alpha\in(0,1/2)$ such that $\alpha>\beta$ and we define the events
$B^{1}_{n}$ and $B^{2}_{n}$ respectively as

$\displaystyle B^{1}_{n}$
$\displaystyle:=\\{\sigma_{1}=\sigma_{1}({\mathbf{w}}_{1},{\mathbf{w}}_{2})<2n^{\beta}\\}\text{
and }$ $\displaystyle B^{2}_{n}$
$\displaystyle:=\cap_{i=1}^{2}\bigl{\\{}\max\\{|(h^{j}({\mathbf{w}}_{i})-{\mathbf{w}}_{i})(1)|:1\leq
j\leq\lfloor 2n^{\beta}\rfloor\\}\leq n^{\alpha}\bigr{\\}}.$

The event $B^{1}_{n}$ says that first (joint) renewal step
$\sigma_{1}({\mathbf{w}}_{1},{\mathbf{w}}_{2})$ occurs within the next
$\lfloor 2n^{\beta}\rfloor$ many steps. Recall that we have achieved strong
uniform exponential tail decay for the family
$\\{\tau_{j+1}({\mathbf{w}}_{1},{\mathbf{w}}_{2})-\tau_{j}({\mathbf{w}}_{1},{\mathbf{w}}_{2}):j\geq
0\\}$ through repeated occurrence of $A_{\text{sp}}(\quad)$ events and these
events depend only on the collection
$\\{\Gamma_{\mathbf{w}}:{\mathbf{w}}\in\mathbb{H}^{+}(0)\\}$. Further, given
any realisation of the collection $\\{\Gamma_{\mathbf{w}}:{\mathbf{w}}\in{\cal
R}\\}$ satisfying $\tilde{{\mathbf{w}}}\in\mathbb{H}^{-}(n^{\beta})$ for all
${\mathbf{w}}\in{\cal R}$, probability of the occurrence of ‘Out’ event at a
$\tau_{j}$ step with $\tau_{j}\geq n^{\beta}$ becomes higher than having no
information on $\\{\Gamma_{\mathbf{w}}:{\mathbf{w}}\in{\cal R}\\}$. Therefore,
given any realisation of the collection
$\\{\Gamma_{\mathbf{w}}:{\mathbf{w}}\in{\cal R}\\}$ satisfying
$\tilde{{\mathbf{w}}}\in\mathbb{H}^{-}(n^{\beta})$ for all
${\mathbf{w}}\in{\cal R}$, the argument of Proposition 3.1 still ensures that
${\mathbb{P}}((B^{1}_{n})^{c})$ goes to zero sub-exponentially. On the other
hand, the proof of Item (iv) of Lemma 3.7 uses ‘special’ points in the upper
half-plane $\mathbb{H}^{+}(0)$. Hence, given any realisation of the collection
$\\{\Gamma_{\mathbf{w}}:{\mathbf{w}}\in{\cal R}_{n}\\}$, the probability
${\mathbb{P}}((B^{2}_{n})^{c})$ goes to zero sub-exponentially fast too.

Let $\sigma_{\ell}$ be the first joint renewal step after crossing the line
$y=n^{\beta}$. On the event $B^{1}_{n}\cap B^{2}_{n}$, the coalescing time
$T({\mathbf{w}}_{1},{\mathbf{w}}_{2})$ is dominated by
$T(h^{\sigma_{\ell}}({\mathbf{w}}_{1}),h^{\sigma_{\ell}}({\mathbf{w}}_{2}))+2n^{\beta}$.
We further observe that the coalescing time
$T(h^{\sigma_{\ell}}({\mathbf{w}}_{1}),h^{\sigma_{\ell}}({\mathbf{w}}_{2}))$
depends on the collection of random vectors
$\\{\Gamma_{\mathbf{w}}:{\mathbf{w}}\in\mathbb{H}^{+}(\sigma_{\ell})\\}$ which
is contained in the collection
$\\{\Gamma_{\mathbf{w}}:{\mathbf{w}}\in\mathbb{H}^{+}(n^{\beta})\\}$.
Therefore, for all large $n$ we obtain

$\displaystyle{\mathbb{P}}\Bigl{(}T({\mathbf{w}}_{1},{\mathbf{w}}_{2})>n\mid\bigl{(}\\{\Gamma_{\mathbf{w}}:{\mathbf{w}}\in{\cal
R}\\}\text{ such that }\tilde{{\mathbf{w}}}\in\mathbb{H}^{-}(n^{\beta})\text{
for all }{\mathbf{w}}\in{\cal R}\bigr{)}\Bigr{)}$
$\displaystyle\leq{\mathbb{P}}\Bigl{(}(T({\mathbf{w}}_{1},{\mathbf{w}}_{2})>n)\cap
B^{1}_{n}\cap B^{2}_{n}\mid\bigl{(}\\{\Gamma_{\mathbf{w}}:{\mathbf{w}}\in{\cal
R}\\}\text{ such that }\tilde{{\mathbf{w}}}\in\mathbb{H}^{-}(n^{\beta})\text{
for all }{\mathbf{w}}\in{\cal R}\bigr{)}\Bigr{)}$
$\displaystyle\qquad\qquad\qquad+{\mathbb{P}}((B^{1}_{n})^{c})+{\mathbb{P}}((B^{2}_{n})^{c})$
$\displaystyle\leq{\mathbb{P}}\Bigl{(}(T(h^{\sigma_{1}}({\mathbf{w}}_{1}),h^{\sigma_{1}}({\mathbf{w}}_{2}))+2n^{\beta}>n)\cap
B^{1}_{n}\cap B^{2}_{n}\mid\bigl{(}\\{\Gamma_{\mathbf{w}}:{\mathbf{w}}\in{\cal
R}\\}\text{ such that }$
$\displaystyle\qquad\qquad\tilde{{\mathbf{w}}}\in\mathbb{H}^{-}(n^{\beta})\text{
for all }{\mathbf{w}}\in{\cal
R}\bigr{)}\Bigr{)}+{\mathbb{P}}((B^{1}_{n})^{c})+{\mathbb{P}}((B^{2}_{n})^{c})$
$\displaystyle\leq{\mathbb{P}}\Bigl{(}(T(h^{\sigma_{1}}({\mathbf{w}}_{1}),h^{\sigma_{1}}({\mathbf{w}}_{2}))>n/2)\cap
B^{1}_{n}\cap B^{2}_{n}\mid\bigl{(}\\{\Gamma_{\mathbf{w}}:{\mathbf{w}}\in{\cal
R}\\}\text{ such that }$
$\displaystyle\qquad\qquad\tilde{{\mathbf{w}}}\in\mathbb{H}^{-}(n^{\beta})\text{
for all }{\mathbf{w}}\in{\cal
R}\bigr{)}\Bigr{)}+{\mathbb{P}}((B^{1}_{n})^{c})+{\mathbb{P}}((B^{2}_{n})^{c})$
$\displaystyle={\mathbb{P}}\Bigl{(}(T(h^{\sigma_{1}}({\mathbf{w}}_{1}),h^{\sigma_{1}}({\mathbf{w}}_{2}))>n/2)\cap
B^{1}_{n}\cap
B^{2}_{n}\Bigr{)}+{\mathbb{P}}((B^{1}_{n})^{c})+{\mathbb{P}}((B^{2}_{n})^{c})$
$\displaystyle\leq
C_{0}({\mathbf{w}}_{2}(1)-{\mathbf{w}}_{1}(1)+2n^{\alpha})/\sqrt{n}$
$\displaystyle\leq
C^{\prime}_{0}({\mathbf{w}}_{2}(1)-{\mathbf{w}}_{1}(1))/\sqrt{n},$

for some $C^{\prime}_{0}>0$. This completes the proof. ∎

We are ready to prove Theorem 1.1 now.

Proof of Theorem 1.1: We prove that the PH network $G$ is a.s. connected. We
prove the second part of the Theorem 1.1, i.e., there is no bi-infinite path
a.s. later in Remark 7.4.

Theorem 6.1 gives us that the coalescing time of two PH paths starting from
${\mathbf{w}}_{1},{\mathbf{w}}_{2}\in{\mathbb{Z}}^{2}$ with
${\mathbf{w}}_{1}(2)={\mathbf{w}}_{2}(2)$ s finite a.s. Therefore, we have
that

$\displaystyle{\mathbb{P}}\bigl{[}\bigcap_{{\mathbf{w}}_{1},{\mathbf{w}}_{2}\in{\mathbb{Z}}^{2},{\mathbf{w}}_{1}(2)={\mathbf{w}}_{2}(2)}\\{\text{
the paths }\pi^{{\mathbf{w}}_{1}},\pi^{{\mathbf{w}}_{2}}\text{ coalesce
eventually}\\}\bigr{]}=1.$ (54)

Because of the non-crossing nature of our model, it is straightforward to
observe that (54) proves the first part of Theorem 1.1. ∎

##  7 Convergence to the Brownian web

This section is devoted to the proof of Theorem 1.2, i.e, the collection of
scaled PH paths converges to the Brownian web (BW). In fact we prove a
stronger version of the theorem in the sense that we construct a dual process
and show that under diffusive scaling the original process together with the
dual process jointly converge to the BW and its dual. Towards this we will
apply a robust technique that was developed in [CSST20] to study convergence
to the BW for non-crossing path models. We recall here that the PH paths are
non-crossing in the sense of (7).

We recall that the BW and its dual denoted by
$({\mathcal{W}},\widehat{{\mathcal{W}}})$ is a
$({\mathcal{H}}\times\widehat{{\mathcal{H}}},{\mathcal{B}}_{{\mathcal{H}}\times\widehat{{\mathcal{H}}}})$-valued
random variable such that:

  * $(i)$

$\widehat{{\mathcal{W}}}$ is distributed as $-{\mathcal{W}}$, the BW rotated
$180^{0}$ about the origin;

  * $(ii)$

${\mathcal{W}}$ and $\widehat{{\mathcal{W}}}$ uniquely determine each other in
the sense that the paths of ${\cal W}$ a.s. do not cross with (backward) paths
in $\widehat{{\cal W}}$. See Theorem 2.4 of Schertzer et al. [SSS19]. The
interaction between paths in ${\mathcal{W}}$ and $\widehat{\mathcal{W}}$ is
that of Skorohod reflection (see [STW00]).

It is time to specify a dual graph $\widehat{G}$ for the PH network $G$. The
construction of the dual graph is not unique and our construction is inspired
from that of [RSS16B]. For our dual graph $\widehat{G}$, the dual vertices are
precisely the mid-points between two consecutive vertices in $V$ on each
horizontal line $y=m$ for $m\in{\mathbb{Z}}$. It is further ensured that each
dual vertex has a unique offspring dual vertex in the negative direction of
the y-axis. Before giving a formal definition, we direct the attention of the
reader to Figure 6.

[subgriddiv=0,griddots=10,gridlabels=0]  Figure 6: The shaded (black or gray)
points are points of $V$ and the black arrows represent the PH paths. Black
points are special points whereas gray points general points of $V$, i.e.,
they are perturbed versions of some other open lattice points. The white
circled points are the points of the dual process and the gray (dashed) paths
are the dual paths.

For $(x,t)\in{\mathbb{Z}}^{2}$, we define,

$\begin{split}J^{+}_{(x,t)}&:=\inf\\{k:k\geq 1,(x+k,t)\in V\\}\\\
J^{-}_{(x,t)}&:=\inf\\{k:k\geq 1,(x-k,t)\in V\\}.\end{split}$ (55)

Next, we define $r(x,t):=(x+J^{+}_{(x,t)},t)$ and
$l(x,t):=(x-J^{+}_{(x,t)},t)$, as the first open point to the right (open
right neighbour) and the first open point to the left (open left neighbour) of
$(x,t)$ at the same level $y=t$ respectively. For $(x,t)\in V$, let
$\widehat{r}(x,t):=(x+J^{+}_{(x,t)}/2,t)$ and
$\widehat{l}(x,t):=(x-J^{-}_{(x,t)}/2,t)$ respectively denote the right dual
neighbour and the left dual neighbour of $(x,t)$ in the dual vertex set.
Finally, the dual vertex set is given by

$\widehat{V}:=\\{\widehat{r}(x,t),\widehat{l}(x,t):(x,t)\in V\\}.$

For a vertex $(u,s)\in\widehat{V}$, let $(v,s-1)\in\widehat{V}$ be such that
the straight line segment joining $(u,s)$ and $(v,s-1)$ does not cross any
edge in $G$. The dual edges are edges joining all such $(u,s)$ and $(v,s-1)$.
Formally, for $(u,s)\in\widehat{V}$, we define

$\begin{split}a^{l}(u,s)&:=\sup\\{z:(z,s-1)\in V,h(z,s-1)(1)<u\\}\\\
a^{r}(u,s)&:=\inf\\{z:(z,s-1)\in V,h(z,s-1)(1)>u\\}\end{split}$ (56)

and set $\widehat{h}(u,s):=((a^{l}(u,s)+a^{r}(u,s))/2,s-1)$. Note that
$(a^{r}(u,s),s-1)$ and $(a^{l}(u,s),s-1)$ are the nearest vertices in $V$ to
the right and left respectively of the dual vertex $\widehat{h}(u,s)$. Finally
the edge set of the dual graph $\widehat{G}:=(\widehat{V},\widehat{E})$ is
given by

$\widehat{E}:=\\{\langle(u,s),\widehat{h}(u,s)\rangle:(u,s)\in\widehat{V}\\}.$

Clearly, each dual vertex has exactly one outgoing edge which goes in the
downward direction. Hence, the dual graph
$\widehat{G}:=(\widehat{V},\widehat{E})$ does not contain any cycle or loop.
This forest $\widehat{G}$ is entirely determined from $G$ without any extra
randomness.

The dual (or backward) path $\widehat{\pi}^{(y,s)}\in\widehat{\Pi}$ starting
at $(y,s)$ is constructed by linearly joining the successive
$\widehat{h}(\cdot)$ steps. Thus, $\widehat{{\cal
X}}:=\\{\widehat{\pi}^{(y,s)}:(y,s)\in\widehat{V}\\}$ denotes the collection
of all dual paths obtained from $\widehat{G}$.

Let us recall that ${\cal X}_{n}={\cal X}_{n}(\gamma,\sigma)$ for
$\gamma,\sigma>0$ and $n\geq 1$, is the collection of $n$-th order diffusively
scaled paths. In the same way, we define $\widehat{{\cal
X}}_{n}=\widehat{{\cal X}}_{n}(\gamma,\sigma)$ as the collection of
diffusively scaled dual paths. For any dual path $\widehat{\pi}$ with starting
time $\sigma_{\widehat{\pi}}$, the scaled dual path
$\widehat{\pi}_{n}(\gamma,\sigma):[-\infty,\sigma_{\widehat{\pi}}/n\gamma]\to[-\infty,\infty]$
is given by

$\widehat{\pi}_{n}(\gamma,\sigma)(t):=\widehat{\pi}(n\gamma
t)/\sqrt{n}\sigma~{}.$ (57)

For each $n\geq 1$, the closure of $\widehat{{\cal X}}_{n}$ in
$(\widehat{\Pi},d_{\widehat{\Pi}})$ denoted as $\overline{\widehat{{\cal
X}}}_{n}$ is a $(\widehat{{\cal H}},{\cal B}_{\widehat{{\cal H}}})$-valued
random variable. We are now ready to state our result regarding joint
convergence for PH model:

######  Theorem 7.1.

There exist $\sigma=\sigma(p,\theta_{x},\theta_{y})>0$ and
$\gamma=\gamma((p,\theta_{x},\theta_{y}))>0$ such that the sequence

$\big{\\{}\big{(}\overline{{\cal
X}}_{n}(\gamma,\sigma),\overline{\widehat{\cal
X}}_{n}(\gamma,\sigma)\big{)}:\,n\geq 1\big{\\}}$

converges in distribution to $({\cal W},\widehat{{\cal W}})$ as $({\cal
H}\times\widehat{{\cal H}},{\cal B}_{{\cal H}\times\widehat{{\cal
H}}})$-valued random variables as $n\rightarrow\infty$.

The convergence criteria to the BW for non-crossing path models are provided
by Fontes et al. [FINR04]. Schertzer et al. [SSS19] provides a very complete
overview on this topic. Let $\Xi\subset\Pi$. For $t>0$ and
$t_{0},a,b\in{\mathbb{R}}$ with $a<b$, let $\eta_{\Xi}(t_{0},t;a,b)$ denote
the counting random variable defined as

$\eta_{\Xi}(t_{0},t;a,b):=\\#\big{\\{}\pi(t_{0}+t):\,\pi\in\Xi,\,\sigma_{\pi}\leq
t_{0}\text{ and }\pi(t_{0})\in[a,b]\big{\\}}.$ (58)

In other words, $\eta_{\Xi}(t_{0},t;a,b)$ considers all paths in $\Xi$, born
before $t_{0}$, that intersect $[a,b]$ at time $t_{0}$ and counts the number
of different positions these paths occupy at time $t_{0}+t$. In Theorem 2.2 of
[FINR04], Fontes et al. provided the following convergence criteria.

######  Theorem 7.2 (Theorem 2.2 of [FINR04]).

Let $\\{\Xi_{n}:n\in{\mathbb{N}}\\}$ be a sequence of
$({\mathcal{H}},B_{{\mathcal{H}}})$ valued random variables with non-crossing
paths. Assume that the following conditions hold:

  * $(I_{1})$

Fix a deterministic countable dense set ${\mathcal{D}}$ of ${\mathbb{R}}^{2}$.
For each ${\mathbf{x}}\in{\mathcal{D}}$, there exists
$\pi_{n}^{{\mathbf{x}}}\in\Xi_{n}$ such that for any finite set of points
${\mathbf{x}}^{1},\dotsc,{\mathbf{x}}^{k}\in{\mathcal{D}}$, as $n\to\infty$,
we have $(\pi^{{\mathbf{x}}^{1}}_{n},\dotsc,\pi^{{\mathbf{x}}^{k}}_{n})$
converges in distribution to
$(W^{{\mathbf{x}}^{1}},\dotsc,W^{{\mathbf{x}}^{k}})$, where
$(W^{{\mathbf{x}}^{1}},\dotsc,W^{{\mathbf{x}}^{k}})$ denotes coalescing
Brownian motions starting from the points
${\mathbf{x}}_{1},\ldots,{\mathbf{x}}_{k}$.

  * $(B_{1})$

For all $t>0$,
$\limsup_{n\rightarrow\infty}\sup_{(a,t_{0})\in{\mathbb{R}}^{2}}{\mathbb{P}}(\eta_{\Xi_{n}}(t_{0},t;a,a+\epsilon)\geq
2)\rightarrow 0$ as $\epsilon\downarrow 0$.

  * $(B_{2})$

For all $t>0$,
$\frac{1}{\epsilon}\limsup_{n\rightarrow\infty}\sup_{(a,t_{0})\in{\mathbb{R}}^{2}}{\mathbb{P}}(\eta_{\Xi_{n}}(t_{0},t;a,a+\epsilon)\geq
3)\rightarrow 0$ as $\epsilon\downarrow 0$.

Then $\Xi_{n}$ converges in distribution to the standard Brownian web
${\mathcal{W}}$ as $n\to\infty$.

Let us first mention that for a sequence of $({\cal H},{\cal B}_{{\cal
H}})$-valued random variables $\\{\Xi_{n}:n\in{\mathbb{N}}\\}$ with non-
crossing paths, Criterion $(I_{1})$ implies tightness (see Proposition B.2 in
the Appendix of [FINR04] or Proposition 6.4 in [SSS19]) and hence sub-
sequential limit(s) always exists. Moreover, Criterion $(B_{1})$ has been
shown to be redundant with $(I_{1})$ for non-crossing path models (see Theorem
6.5 of [SSS19]). Actually Condition $(I_{1})$ implies that subsequential limit
contains coalescing Brownian motions starting from all points with rational
coordinates and hence contain a copy of the standard BW ${\mathcal{W}}$.
Through Condition $(B_{2})$, we ensure that the limiting random variable does
not have extra paths other than the BW.

Criterion $(B_{2})$ is often verified by applying an FKG type correlation
inequality together with an estimate on the distribution of the coalescence
time between two paths. However, FKG is a strong property and difficult to
apply for models with complicate dependencies. We will follow a more robust
technique developed in [CSST20] and applicable only for non-crossing path
models. The following theorem is taken from [CSST20] to obtain joint
convergence for the PH and its dual to the BW and its dual.

######  Theorem 7.3 (Theorem 6.3 of [CSST20]).

Let $\\{(\Xi_{n},\widehat{\Xi}_{n}):n\geq 1\\}$ be a sequence of $({\cal
H}\times\widehat{{\cal H}},{\cal B}_{{\cal H}\times\widehat{{\cal
H}}})$-valued random variables with non-crossing paths only, satisfying the
following assumptions:

  * (i)

For each $n\geq 1$, paths in $\Xi_{n}$ do not cross (backward) paths in
$\widehat{\Xi}_{n}$ almost surely, i.e., there does not exist any
$\pi\in\Xi_{n}$, $\widehat{\pi}\in\widehat{\Xi}_{n}$ and
$t_{1},t_{2}\in(\sigma_{\pi},\sigma_{\widehat{\pi}})$ such that
$(\widehat{\pi}(t_{1})-\pi(t_{1}))(\widehat{\pi}(t_{2})-\pi(t_{2}))<0$ almost
surely.

  * (ii)

$\\{\Xi_{n}:n\in{\mathbb{N}}\\}$ satisfies $(I_{1})$.

  * (iii)

$\\{(\widehat{\pi}_{n}(\sigma_{\widehat{\pi}_{n}}),\sigma_{\widehat{\pi}_{n}}):\widehat{\pi}_{n}\in\widehat{\Xi}_{n}\\}$,
the collection of starting points of all the backward paths in
$\widehat{\Xi}_{n}$, as $n\to\infty$, becomes dense in ${\mathbb{R}}^{2}$.

  * (iv)

For any sub sequential limit $({\cal Z},\widehat{{\cal Z}})$ of
$\\{(\Xi_{n},\widehat{\Xi}_{n}):n\in{\mathbb{N}}\\}$, paths of ${\cal Z}$ do
not spend positive Lebesgue measure time together with paths of
$\widehat{{\cal Z}}$, i.e., almost surely there is no $\pi\in{\cal Z}$ and
$\widehat{\pi}\in\widehat{{\cal Z}}$ such that
$\int_{\sigma_{\pi}}^{\sigma_{\widehat{\pi}}}\mathbf{1}_{\pi(t)=\widehat{\pi}(t)}dt>0$.

Then $({\cal X}_{n},\widehat{{\cal X}}_{n})$ converges in distribution $({\cal
W},\widehat{{\cal W}})$ as $n\to\infty$.

It is useful to mention here that there are several other approaches to
replace Criterion $(B_{2})$. Long before, Criterion $(E)$ was proposed by
Newman et al [NRS05] which is applicable even for models with crossing paths
as well. Schertzer et al. [SSS19] provided a new criterion in Theorem 6.6
replacing $(B_{2})$, called the wedge condition. Theorem 7.3 appears as a
slight generalization of Theorem 6.6 of [SSS19] by considering the joint
convergence to the BW and it’s dual. It replaces the wedge condition by the
fact that no limiting primal and dual paths can spend positive Lebesgue time
together. The next subsection is devoted to verification of the conditions of
Theorem 7.3 for the diffusively scaled PH and its dual $\\{(\overline{{\cal
X}}_{n},\overline{\widehat{{\cal X}}}_{n}):n\in{\mathbb{N}}\\}$.

Before proceeding further, we make the following remark regarding existence of
bi-infinite path for the PH network and this completes the proof of Theorem
1.1. We comment here that it is possible to obtain the same result by
following a Burton-Keane argument.

######  Remark 7.4.

From the construction of the dual graph it is evident that the PH network has
a bi-infinite path if and only if the dual graph is not connected. If there
are scaled dual paths which do not coalesce but converge to coalescing
Brownian motions then there must be scaled forward paths entrapped between
these scaled dual paths. Further, joint convergence to the double Brownian web
$({\mathcal{W}},\widehat{{\mathcal{W}}})$ forces that there must be a limiting
forward Brownian path approximating this sequence of entrapped forward scaled
paths and this limiting forward Brownian path must spend positive Lebesgue
measure time together with a backward (dual) Brownian path. This leads to a
contradiction and gives us that there is no bi-infinite path in the PH network
a.s.

###  7.1 Verification of conditions of Theorem 7.3

In this section, we show that the sequence of diffusively scaled path families
$\\{(\overline{{\cal X}}_{n},\overline{\widehat{{\cal X}}}_{n}):n\geq 1\\}$
obtained from the PH network and its dual satisfies the conditions in Theorem
7.3.

Conditions $(i)$ and $(iii)$ of Theorem 7.3 hold by construction. Indeed,
paths of ${\cal X}$ do not cross (backward) paths of $\widehat{{\cal X}}$ with
probability $1$ and the same holds for $\overline{{\cal X}}_{n}$ and
$\overline{\widehat{{\cal X}}}_{n}$ for any $n\geq 1$. Clearly the set of all
starting points of the scaled backward paths in $\widehat{\Xi}_{n}$ becomes
dense in ${\mathbb{R}}^{2}$ as $n\to\infty$. We prove Condition (ii) in the
following subsection.

####  7.1.1 Verification of Condition $(I_{1})$

In this section we show that the condition $(ii)$ holds for the sequence
$\\{\overline{{\cal X}}_{n}:n\geq 1\\}$, i.e., Criterion $(I_{1})$ of Theorem
7.2 holds.

We first focus on a single path, $\pi^{\mathbf{0}}$ starting at the origin.
The main ingredient here is the construction of i.i.d. pieces through
(marginal) renewal steps. As shown in Proposition 4.1, the sequence of renewal
steps $\\{h^{\sigma_{\ell}}(\mathbf{0})(1):\ell\geq 1\\}$ breaks down the path
$\pi^{\mathbf{0}}$ into independent pieces. Let us scale the PH path
$\pi^{\mathbf{0}}$ starting from $\mathbf{0}$ into $\pi^{\mathbf{0}}_{n}$ as
in (6) with the following choices of $\sigma$ and $\gamma$

$\displaystyle\sigma^{2}$
$\displaystyle:=\text{Var}(Y_{2}(1)-Y_{1}(1))=\text{Var}\bigl{(}(h^{\sigma_{2}}(\mathbf{0})-h^{\sigma_{1}}(\mathbf{0}))(1)\bigr{)}\text{
and }$ $\displaystyle\gamma$
$\displaystyle:={\mathbb{E}}(Y_{2}(2)-Y_{1}(2))={\mathbb{E}}\bigl{(}h^{\sigma_{2}}(\mathbf{0})(2)-h^{\sigma_{1}}(\mathbf{0})(2)\bigr{)}.$
(59)

From now on, the diffusively scaled sequence $\\{\overline{{\cal X}}_{n}:n\geq
1\\}$ is considered w.r.t. these parameters, but for ease of writing, we drop
$(\gamma,\sigma)$ from our notation. Proposition 4.1 together with Corollary
4.5 allow us to apply Donsker’s invariance principle to show that
$\pi^{\mathbf{0}}_{n}$ converges in distribution in $(\Pi,d_{\Pi})$ to
$B^{\mathbf{0}}$, a standard Brownian motion starting at $\mathbf{0}$.

The above argument proves Criterion $(I_{1})$ for $k=1$. To prove Criterion
$(I_{1})$ for general $k\geq 1$ we follow the method of induction. We proceed
to prove it for $k\geq 2$ assuming it is true for $k-1$. The strategy that we
adopt is to show that until the time when the $k$-th PH path comes close to
one of the other $(k-1)$ PH paths, it can be approximated by an independent
path with the same distribution as itself, and after that time, it quickly
coalesces with the path which is close to it and both of them converge to the
same Brownian motion. Proposition 6.3, which has been derived from Theorem
6.1, allows us to show that when two PH paths come close enough, after that
time they coalesce quickly. This strategy was first developed by Ferrari et al
in [FFW05] to deal with dependent paths with bounded range interactions, and
later modified in a series of papers ([CFD09], [RSS16A], [CSST20]) to deal
with long range interactions. Since, the essential idea of the proof here is
the same as that of [CSST20], we do not provide the full details and refer the
reader to Section 6.2.1 of [CSST20]. Still, because of complicate dependencies
of our model, we need to make some adjustments.

Thus we obtain that, for ${\mathbf{v}}\in{\mathbb{R}}^{2}$ and for any
sequence ${\mathbf{v}}^{n}\in{\mathbb{Z}}^{2}$ such that
$({\mathbf{v}}^{n}(1)/\sqrt{n}\sigma,{\mathbf{v}}^{n}(2)/n\gamma)\to{\mathbf{v}}$
as $n\to\infty$, the scaled PH path $\pi^{{\mathbf{v}}^{n}}_{n}$ converges in
distribution to $B^{\mathbf{v}}$, Brownian motion starting from
${\mathbf{v}}$. In fact, non-crossing property of paths imply that for
sequences ${\mathbf{v}}^{n}$ and ${\mathbf{w}}_{n}$ in ${\mathbb{Z}}^{2}$ such
that ${\mathbf{v}}^{n}(2)={\mathbf{w}}^{n}(2)=0$,
${\mathbf{w}}^{n}(1)<0<{\mathbf{v}}^{n}(1)$ with
$({\mathbf{v}}^{n}(1)-{\mathbf{w}}^{n}(1))/\sqrt{n}\to 0$, the couple
$(\pi^{{\mathbf{w}}^{n}}_{n},\pi^{{\mathbf{v}}^{n}}_{n})$ converges in
distribution to $(B^{\mathbf{0}},B^{\mathbf{0}})$, i.e., to the same Brownian
motion starting from origin. This is also implied by the estimated on the
coalescing time that we have established in Theorem 6.1. This completes the
proof of Criterion $(I_{1})$ for $k=1$.

We now define a sequence of subsets of $\Pi^{k}$ where the $k$ th path comes
close to one of the $k-1$ paths. We fix $\alpha\in(0,1/2)$ for the rest of
this section. For $n\geq 1$, define

$\displaystyle A_{n}^{\alpha}=$
$\displaystyle\Bigl{\\{}(\pi_{1},\dotsc,\pi_{k})\in\Pi^{k}:\pi_{i}\text{'s
satisfy the following conditions }$ $\displaystyle
a)\quad\pi_{i}(\sigma_{\pi_{i}})\neq\pi_{j}(\sigma_{\pi_{j}})\text{ for all
}1\leq i<j\leq k;$ $\displaystyle b)\quad t^{k}_{n}:=\inf\\{t\geq
0:|\pi_{i}(t)-\pi_{k}(t)|\leq 3n^{\alpha-1/2}\text{ for some }1\leq i\leq
k-1\\}<\infty\Bigr{\\}}.$ (60)

Next we define the ‘$\alpha$-coalescence map’
$f_{n}^{(\alpha)}:\Pi^{k}\to\Pi^{k}$, which is actually a modification of the
coalescence map $f_{n}$ introduced in Ferrari et al. [FFW05], as follows:

$f_{n}^{(\alpha)}(\pi_{1},\dotsc,\pi_{k}):=\begin{cases}(\pi_{1},\dotsc,\pi_{k-1},\overline{\pi}_{k})&\text{for
}(\pi_{1},\dotsc,\pi_{k})\in A_{n}^{\alpha}\\\ (\pi_{1},\dotsc,\pi_{k})&\text{
otherwise }\end{cases}$

with

$\displaystyle\overline{\pi}_{k}(t):=\begin{cases}\pi_{k}(t)&\text{ for }t\leq
t^{k}_{n}\\\
\pi_{k}(t^{k}_{n})+n(t-t^{k}_{n})\bigl{[}\pi_{i}(t^{k}_{n}+\frac{1}{n})-\pi_{k}(t^{k}_{n})\bigr{]}&\text{
for }t^{k}_{n}<t\leq t^{k}_{n}+\frac{1}{n}\\\ \overline{\pi}_{i}(t)&\text{ for
}t>t^{k}_{n}+\frac{1}{n}\end{cases}$

where $i$ is the index such that $|\pi_{i}(t^{k}_{n})-\pi_{k}(t^{k}_{n})|\leq
3n^{\alpha-1/2}$ and $|\pi_{j}(t^{k}_{n})-\pi_{k}(t^{k}_{n})|>3n^{\alpha-1/2}$
for all $1\leq j<i$.

Fix any set of $k$ points ${\mathbf{y}}_{1},\cdots,{\mathbf{y}}_{k}$ in
${\mathbb{R}}^{2}$ and let
$(W^{{\mathbf{y}}_{1}},\dotsc,W^{{\mathbf{y}}_{k}})$ denote coalescing
Brownian motions starting from these points. Let
$\\{({\mathbf{y}}^{n}_{1},\cdots,{\mathbf{y}}^{n}_{k}):n\in{\mathbb{N}}\\}$ be
such that for all $1\leq i\leq k$ we have

$({\mathbf{y}}^{n}_{i}(1)/\sqrt{n}\sigma,{\mathbf{y}}^{n}_{i}(1)/n\gamma)\to{\mathbf{y}}_{i}\text{
as }n\to\infty.$

Let $\pi^{i}$ denote the PH path starting from ${\mathbf{y}}^{n}_{i}$ and
$\pi^{i}_{n}$ represents the $n$-th diffusively scaled version. The following
proposition completes the verification of condition $(I_{1})$.

######  Proposition 7.5.

We have, as $n\to\infty$,

$\displaystyle(a)~{}$ $\displaystyle
f^{(\alpha)}_{n}(\pi^{1}_{n},\dotsc,\pi^{k-1}_{n},\pi^{k}_{n})\Rightarrow(W^{{\mathbf{y}}_{1}},\dotsc,W^{{\mathbf{y}}_{k}});$
$\displaystyle(b)~{}$
$\displaystyle(\pi^{1}_{n},\dotsc,\pi^{k-1}_{n},\pi^{k}_{n})\Rightarrow(W^{{\mathbf{y}}_{1}},\dotsc,W^{{\mathbf{y}}_{k}}).$

###### Proof.

For $\pi\in\Pi$ and $t>\sigma_{\pi}$ let
$\pi_{[\sigma_{\pi},t]}=\pi_{(-\infty,t]}$ denotes the restriction of $\pi$
over the time interval $[\sigma_{\pi},t]$. For
$(\pi_{1},\cdots,\pi_{k})\in\Pi^{k}$ and for $t>\max\\{\sigma_{\pi_{i}}:1\leq
i\leq k\\}$ the notation $(\pi_{1},\cdots,\pi_{k})_{(-\infty,t]}$ denotes the
tuple of respective restrictions of individual paths. Fix
$t>\max\\{{\mathbf{y}}_{i}(2):1\leq i\leq k\\}$ and we observe that for Item
(a) it is enough to show that

$\displaystyle
f^{(\alpha)}_{n}((\pi^{1}_{n},\dotsc,\pi^{k-1}_{n},\pi^{k}_{n})_{(-\infty,t]})\Rightarrow(W^{{\mathbf{y}}_{1}},\dotsc,W^{{\mathbf{y}}_{k}})_{(-\infty,t]}.$
(61)

Further, in order to prove (61) it suffices to show that

$\displaystyle{\mathbb{E}}\bigl{[}g\bigl{(}f^{(\alpha)}_{n}((\pi^{1}_{n},\dotsc,\pi^{k-1}_{n},\pi^{k}_{n})_{(-\infty,t]})\bigr{)}\bigr{]}\Rightarrow{\mathbb{E}}\bigl{[}g\bigl{(}(W^{{\mathbf{y}}_{1}},\dotsc,W^{{\mathbf{y}}_{k}})_{(-\infty,t]}\bigr{)}\bigr{]},$
(62)

for bounded continuous function $g$. We need to introduce some notations. Our
motivation is to show that with high probability a PH path uses perturbed
versions of ‘close’ enough lattice points only and therefore, as long as two
PH paths are far away, they evolve independently.

Fix $\beta\in(0,\alpha)$ where $\alpha$ is as in (37). We consider ‘Tubes’ of
width $n^{\alpha}$ and $n^{\beta}$ around each of the $k$ (unscaled) PH paths
as defined below:

$\displaystyle\mathbb{T}^{\alpha}_{n,i}$
$\displaystyle:=\\{(y,s)\in{\mathbb{Z}}^{2}:||(y,s)-h^{m}({\mathbf{y}}^{n}_{i})||_{1}\leq
n^{\alpha}\text{ for some }0\leq m\leq\lfloor n\gamma
t-{\mathbf{y}}^{n}_{i}(2)\rfloor\\}\text{ and }$
$\displaystyle\mathbb{T}^{\beta}_{n,i}$
$\displaystyle:=\\{(y,s)\in{\mathbb{Z}}^{2}:||(y,s)-h^{m}({\mathbf{y}}^{n}_{i})||_{1}\leq
n^{\beta}\text{ for some }0\leq m\leq\lfloor n\gamma
t-{\mathbf{y}}^{n}_{i}(2)\rfloor\\}$

where $t$ is as in (61). Let the event $E_{n}$ is defined as bellow

$\displaystyle E_{n}:=\bigcap_{i=1}^{k}$ $\displaystyle\bigl{\\{}\text{There
does not exist any }{\mathbf{w}}\text{ in
}{\mathbb{Z}}^{2}\setminus\mathbb{T}^{\alpha}_{n,i}\text{ with
}\tilde{{\mathbf{w}}}\in\mathbb{T}^{\beta}_{n,i}\bigr{\\}}.$

Exponential tail decay of perturbation random vectors gives us that
${\mathbb{P}}(E_{n})$ converges to $1$ as $n\to\infty$. Therefore, we can
modify (61) as

$\displaystyle{\mathbb{E}}\bigl{[}g\bigl{(}f^{(\alpha)}_{n}((\pi^{1}_{n},\dotsc,\pi^{k-1}_{n},\pi^{k}_{n})_{(-\infty,t]})\bigr{)}\mathbf{1}_{E_{n}}\bigr{]}\to{\mathbb{E}}\bigl{[}g\bigl{(}(W^{{\mathbf{y}}_{1}},\dotsc,W^{{\mathbf{y}}_{k}})_{(-\infty,t]}\bigr{)}\bigr{]}\text{
as }n\to\infty.$ (63)

For each $n\geq 1$ let $\overline{\pi}^{k}_{n}$ denote a scaled PH path which
has the same marginal distribution as that of $\pi^{k}_{n}$ and evolves
independently of $(\pi^{1}_{n},\cdots,\pi^{k-1}_{n})$. A simple construction
using another i.i.d. collection of random vectors
$\\{\Gamma^{\text{ind}}_{w}:{\mathbf{w}}\in{\mathbb{Z}}^{2}\\}$ independent of
the collection $\\{\Gamma_{w}:{\mathbf{w}}\in{\mathbb{Z}}^{2}\\}$ ensures that
such a random path $\overline{\pi}^{k}_{n}$ exists.

We observe that on the event $E_{n}$, till time $t\wedge t^{k}_{n}$, the path
$\pi^{k}_{n}$ is supported on disjoint sets of random vectors and hence, it’s
evolution is independent of all of the $k-1$ paths
$\pi^{1}_{n},\cdots,\pi^{k-1}_{n}$. Therefore, we have

${\mathbb{E}}\bigl{[}g\bigl{(}f^{(\alpha)}_{n}((\pi^{1}_{n},\dotsc,\pi^{k-1}_{n},\pi^{k}_{n})_{(-\infty,t]})\bigr{)}\mathbf{1}_{E_{n}}\bigr{]}\stackrel{{\scriptstyle
d}}{{=}}{\mathbb{E}}\bigl{[}g\bigl{(}f^{(\alpha)}_{n}((\pi^{1}_{n},\dotsc,\pi^{k-1}_{n},\overline{\pi}^{k}_{n})_{(-\infty,t]})\bigr{)}\mathbf{1}_{E_{n}}\bigr{]}.$

The same argument as in Item (a) of Proposition 5.6 in [RSS16A] gives us

${\mathbb{E}}\bigl{[}g\bigl{(}f^{(\alpha)}_{n}((\pi^{1}_{n},\dotsc,\pi^{k-1}_{n},\overline{\pi}^{k-1}_{n})_{(-\infty,t]})\bigr{)}\bigr{]}\to{\mathbb{E}}\bigl{[}g\bigl{(}(W^{{\mathbf{y}}_{1}},\dotsc,W^{{\mathbf{y}}_{k}})_{(-\infty,t]}\bigr{)}\bigr{]}\text{
as }n\to\infty.$

As ${\mathbb{P}}(E_{n})$ converges to $1$ as $n\to\infty$, Equation (63)
follows from the observation that

$\lim_{n\to\infty}{\mathbb{E}}\bigl{[}g\bigl{(}f^{(\alpha)}_{n}(\pi^{1}_{n},\dotsc,\pi^{k-1}_{n},\overline{\pi}^{k}_{n})_{(-\infty,t]})\bigr{)}\mathbf{1}_{E_{n}}\bigr{]}={\mathbb{E}}\bigl{[}g\bigl{(}f^{(\alpha)}_{n}((\pi^{1}_{n},\dotsc,\pi^{k-1}_{n},\overline{\pi}^{k}_{n})_{(-\infty,t]})\bigr{)}\mathbf{1}_{E_{n}}\bigr{]}.$

This proves Item (a).

For (b) we first consider the situation $t^{k}_{n}\geq t$. On the event
$t^{k}_{n}\geq t$, Item (b) follows trivially as we have

$(\pi^{1}_{n},\dotsc,\pi^{k-1}_{n},\pi^{k}_{n})_{(-\infty,t]}=f^{(\alpha)}_{n}(\pi^{1}_{n},\dotsc,\pi^{k-1}_{n},\pi^{k}_{n})_{(-\infty,t]}\text{
a.s.}$

Next, we consider the situation $t>t^{k}_{n}$. W.l.o.g. we assume that at time
$t^{k}_{n}$, paths $\pi^{i}_{n}$ and $\pi^{k}_{n}$ come close enough. We need
to show that after time $t^{k}_{n}$, these two paths quickly coalesce.
Proposition 6.3 would help us to achieve this. Towards that we define the
event

$\displaystyle F_{n}:=\bigcap_{i=1}^{k}$
$\displaystyle\bigl{\\{}\tilde{{\mathbf{w}}}\in\mathbb{H}^{-}(n\gamma
t^{k}_{n}+n^{\beta})\text{ for all
}{\mathbf{w}}\in\mathbb{T}^{\beta}_{n,i}\cap\mathbb{H}^{-}(n\gamma
t^{k}_{n})\bigr{\\}}.$

We observe that ${\mathbb{P}}(F_{n})$ converges to $1$ as $n\to\infty$.
Finally, on the event $F_{n}$, Proposition 6.3 gives us the required tail
estimate for coalescing time of $\pi^{k}_{n}$ and $\pi^{i}_{n}$ and completes
the proof. ∎

####  7.1.2 Verification of condition (iv)

To show condition $(iv)$, we mainly follow Section 6.2.2 of [CSST20] and
again, the coalescence time estimate given in Proposition 6.3 serves as a key
ingredient. Let $({\cal Z},\widehat{\cal Z})$ be any sub-sequential limit of
$\\{(\overline{{\cal X}}_{n},\widehat{\overline{{\cal X}}}_{n}):n\geq 1\\}$.
By Skorohod’s representation theorem we may assume that we are working on a
probability space such that the convergence happens almost surely. With slight
abuse of notation we continue to denote that subsequence also by
$\\{(\overline{{\cal X}}_{n},\widehat{\overline{{\cal X}}}_{n}):n\geq 1\\}$.

We have to prove that, with probability $1$, paths in ${\cal Z}$ do not spend
positive Lebesgue measure time together with the dual paths in $\widehat{\cal
Z}$. This means that for any $\delta>0$ and any integer $m\geq 1$, the
probability of the event

$A(\delta,m):=\left\\{\begin{array}[]{c}\text{$\exists$ paths $\pi\in{\cal
Z},\widehat{\pi}\in\widehat{\cal Z}$ and $t_{0}\in{\mathbb{R}}$ s.t.
$-m<\sigma_{\pi}<t_{0}<t_{0}+\delta<\sigma_{\widehat{\pi}}<m$}\\\ \text{and
$-m<\pi(t)=\widehat{\pi}(t)<m$ for all
$t\in[t_{0},t_{0}+\delta]$}\end{array}\right\\}$

has to be $0$.

To show that ${\mathbb{P}}(A(\delta,m))=0$, we introduce a generic event
$B^{\epsilon}_{n}(\delta,m)$ defined as follows. Given an integer $m\geq 1$
and $\delta,\epsilon>0$,

$\displaystyle B^{\epsilon}_{n}(\delta,m):=\bigl{\\{}$
$\displaystyle\exists\text{ paths }\pi_{1}^{n},\pi_{2}^{n},\pi_{3}^{n}\in{\cal
X}_{n}\text{ s.t.
}\sigma_{\pi_{1}^{n}},\sigma_{\pi_{2}^{n}}=0,\sigma_{\pi_{3}^{n}}\leq\delta\text{
and }\pi_{1}^{n}(0),\pi_{1}^{n}(\delta)\in[-m,m]$ $\displaystyle\text{with
}|\pi_{1}^{n}(0)-\pi_{2}^{n}(0)|<\epsilon\text{ but
}\pi_{1}^{n}(\delta)\not=\pi_{2}^{n}(\delta)$ $\displaystyle\text{and with
}|\pi_{1}^{n}(\delta)-\pi_{3}^{n}(\delta)|<\epsilon\text{ but
}\pi_{1}^{n}(2\delta)\not=\pi_{3}^{n}(2\delta)\bigr{\\}}$

The event $B^{\epsilon}_{n}(\delta,m)$ means that there exists a path
$\pi_{1}^{n}$ localized in $[-m,m]$ at time $0$ as well as at time $\delta$
which is approached (within distance $\epsilon$) by two path $\pi_{2}^{n}$ and
$\pi_{3}^{n}$ respectively at times $0$ and $\delta$ while still being
different from them respectively at time $\delta$ and $2\delta$.

It was shown in Section 6.2.2 of [CSST20] that to show
${\mathbb{P}}(A(\delta,m)=0)$ it suffices to prove the following lemma.

######  Lemma 7.6.

For any integer $m\geq 1$, real numbers $\epsilon,\delta>0$, there exists a
constant $C_{0}(\delta,m)>0$ (only depending on $\delta$ and $m$) such that
for all large $n$,

${\mathbb{P}}(B^{\epsilon}_{n}(\delta,m))\leq C_{0}(\delta,m)\,\epsilon~{}.$

For the proof of Lemma 7.6 we essentially follow [CSST20]. Note that the
discrete nature of the perturbed point process $V$ requires some
modifications. We need to deal with non-Markovian nature of the PH paths.

###### Proof of Lemma 7.6.

We define the event $D^{\epsilon}_{n}$ as the unscaled version of the event
$B^{\epsilon}_{n}$ in the following way:

$\displaystyle D^{\epsilon}_{n}$ $\displaystyle:=\bigl{\\{}\text{there exist
}x,y,z\in{\mathbb{Z}}\text{ such that
}x\in[-m\sqrt{n}\sigma,m\sqrt{n}\sigma],|x-y|<\sqrt{n}\epsilon\sigma\text{ and
}$ $\displaystyle\pi^{(x,0)}(\lfloor
n\gamma\delta\rfloor)\neq\pi^{(y,0)}(\lfloor
n\gamma\delta\rfloor),|\pi^{(x,0)}(\lfloor
n\gamma\delta\rfloor)-z|<\sqrt{n}\epsilon\sigma,\pi^{(x,0)}(2\lfloor
n\gamma\delta\rfloor)\neq\pi^{(z,0)}(2\lfloor
n\gamma\delta\rfloor)\bigr{\\}}.$

For $\omega\in D^{\epsilon}_{n}$, suppose $x,y$ are as in the definition above
and assume that $x<y$. Set $l=\max\\{x+j:\pi^{(x,0)}(\lfloor
n\gamma\delta\rfloor)=\pi^{(x+j,0)}(\lfloor n\gamma\delta\rfloor)\\}$.
Clearly, $-m\sqrt{n}\sigma\leq x\leq l<y\leq(m+\epsilon)\sqrt{n}\sigma$ and
$\pi^{(x,0)}(\lfloor n\gamma\delta\rfloor)=\pi^{(l,0)}(\lfloor
n\gamma\delta\rfloor)<\pi^{(l+1,0)}(\lfloor
n\gamma\delta\rfloor)\leq\pi^{(y,0)}(\lfloor n\gamma\delta\rfloor)$. Assume
that $\pi^{(x,-\delta)}(\lfloor n\gamma\delta\rfloor)=k$ for some
$k\in{\mathbb{Z}}$. Then, $z$ in the definition above satisfies
$z\in(k-\sqrt{n}\epsilon\sigma,k+\sqrt{n}\epsilon\sigma)$ and $\pi^{(k,\lfloor
n\gamma\delta\rfloor)}(2\lfloor n\gamma\delta\rfloor)\neq\pi^{(z,\lfloor
n\gamma\delta\rfloor)}(2\lfloor n\gamma\delta\rfloor)$. So, by non-crossing
property of paths, it must be the case that

$\pi^{(k-\lfloor\sqrt{n}\epsilon\sigma\rfloor-1,\lfloor
n\gamma\delta\rfloor)}(2\lfloor
n\gamma\delta\rfloor)\neq\pi^{(k+\lfloor\sqrt{n}\epsilon\sigma\rfloor+1,\lfloor
n\gamma\delta\rfloor)}(2\lfloor n\gamma\delta\rfloor)$

Thus, we must have $\omega\in H^{(L)}(n,\delta,\epsilon)$ where for
$l\in{\mathbb{Z}}$,

$\displaystyle H_{l,k}^{(L)}(n,\epsilon):=$
$\displaystyle\bigl{\\{}\pi^{(l,0)}(\lfloor
n\gamma\delta\rfloor)=k\neq\pi^{(l+1,0)}(\lfloor n\gamma\delta\rfloor)\text{
and }$ $\displaystyle\pi^{(k-\lfloor\sqrt{n}\epsilon\sigma\rfloor-1,\lfloor
n\gamma\delta\rfloor)}(2\lfloor
n\gamma\delta\rfloor)\neq\pi^{(k+\lfloor\sqrt{n}\epsilon\sigma\rfloor+1,\lfloor
n\gamma\delta\rfloor)}(2\lfloor n\gamma\delta\rfloor)\bigr{\\}};$
$\displaystyle H^{(L)}(n,\delta,\epsilon):=$ $\displaystyle\cup_{l=-\lfloor
2m\sqrt{n}\sigma\rfloor}^{\lfloor
2m\sqrt{n}\sigma\rfloor}\cup_{k\in{\mathbb{Z}}}H_{l,k}^{(L)}(n,\delta,\epsilon).$

Similarly for $\omega\in D^{\epsilon}_{n}$ such that $x>y$, set
$r=\min\\{x-j:\pi^{(x,-\delta)}(\lfloor
n\gamma\delta\rfloor)=\pi^{(x-j,-\delta)}(\lfloor n\gamma\delta\rfloor)\\}$.
As earlier, $\omega\in H^{(R)}(n,\delta,\epsilon)$ where for
$r\in{\mathbb{Z}}$,

$\displaystyle H_{r,k}^{(R)}(n,\delta,\epsilon):=$
$\displaystyle\bigl{\\{}\pi^{(r,0)}(\lfloor
n\gamma\delta\rfloor)=k\neq\pi^{(r-1,0)}(\lfloor n\gamma\delta\rfloor)\text{
and }$ $\displaystyle\pi^{(k-\lfloor\sqrt{n}\epsilon\sigma\rfloor-1,\lfloor
n\gamma\delta\rfloor)}(2\lfloor
n\gamma\delta\rfloor)\neq\pi^{(k+\lfloor\sqrt{n}\epsilon\sigma\rfloor+1,\lfloor
n\gamma\delta\rfloor)}(2\lfloor n\gamma\delta\rfloor)\bigr{\\}};$
$\displaystyle H^{(R)}(n,\delta,\epsilon):=$ $\displaystyle\cup_{r=-\lfloor
2m\sqrt{n}\sigma\rfloor}^{\lfloor
2m\sqrt{n}\sigma\rfloor}\cup_{k\in{\mathbb{Z}}}H_{r,k}^{(R)}(n,\delta,\epsilon).$

Thus, $D^{\epsilon}_{n}\subseteq H^{(L)}(n,\delta,\epsilon)\cup
H^{(R)}(n,\delta,\epsilon)$.

The argument is very similar to Proposition 7.5 and we only give a sketch
here. We consider tubes of widths $n^{\alpha}$ and $n^{\beta}$ around the PH
path $\pi^{(l,0)}$ and $\pi^{(l+1,0)}$ over the time interval $[0,\lfloor
n\gamma\delta\rfloor]$. We observe that the probability of the event that
these two paths do not use random vectors associated to lattice points outside
these ‘$n^{\alpha}$’ tubes to evolve and perturbed versions of all the lattice
points inside these two tubes are confined to the lower half-plane
$\mathbb{H}^{-}(\lfloor n\gamma\delta\rfloor+n^{\beta})$ converges to $1$ as
$n\to\infty$. Therefore for all large $n$, on the said event we can apply
Proposition 6.3 to obtain

$\displaystyle{\mathbb{P}}\\{\pi^{(k-\lfloor\sqrt{n}\epsilon\sigma\rfloor-1,\lfloor
n\gamma\delta\rfloor)}(2\lfloor
n\gamma\delta\rfloor)\neq\pi^{(k+\lfloor\sqrt{n}\epsilon\sigma\rfloor+1,\lfloor
n\gamma\delta\rfloor)}(2\lfloor n\gamma\delta\rfloor)\mid\pi^{(l,0)}(\lfloor
n\gamma\delta\rfloor)=k\neq\pi^{(l+1,0)}(\lfloor n\gamma\delta\rfloor)\\}$
$\displaystyle\leq\frac{C_{2}(2\lfloor\sqrt{n}\sigma\epsilon\rfloor+3)}{\sqrt{\lfloor
n\gamma\delta\rfloor}}\leq C_{3}(\delta)\epsilon$

where $C_{2},C_{3}(\delta)>0$ are constants. Hence,

$\displaystyle{\mathbb{P}}(H_{k}^{(L),1}(n,\delta,\epsilon))\leq
C_{3}(\delta)\epsilon~{}{\mathbb{P}}\\{\pi^{(l,0)}(\lfloor
n\gamma\delta\rfloor)=k\neq\pi^{(l+1,0)}(\lfloor n\gamma\delta\rfloor)\\}.$

Now, the events $\\{\pi^{(l,0)}(\lfloor
n\gamma\delta\rfloor)=k\neq\pi^{(l+1,0)}(\lfloor n\gamma\delta\rfloor)\\}$ are
disjoint for distinct values of $k$. Hence,

$\displaystyle{\mathbb{P}}\bigl{(}\cup_{k\in{\mathbb{Z}}}H_{k}^{(L),1}(n,\delta,\epsilon)\bigr{)}$
$\displaystyle\leq\sum_{k\in{\mathbb{Z}}}{\mathbb{P}}\bigl{(}H_{k}^{(L),1}(n,\delta,\epsilon)\bigr{)}$
$\displaystyle\leq
C_{3}(\delta)\epsilon\sum_{k\in{\mathbb{Z}}}{\mathbb{P}}\\{\pi^{(l,0)}(\lfloor
n\gamma\delta\rfloor)=k\neq\pi^{(l+1,0)}(\lfloor n\gamma\delta\rfloor)\\}$
$\displaystyle=C_{3}(\delta)\epsilon~{}{\mathbb{P}}\\{\pi^{(l,0)}(\lfloor
n\gamma\delta\rfloor)\neq\pi^{(l+1,0)}(\lfloor n\gamma\delta\rfloor)\\}.$

The above argument also holds for
$\cup_{k\in{\mathbb{Z}}}H_{r,k}^{(R)}(n,\delta,\epsilon)$. Thus, combining the
above terms and applying Proposition 6.3

$\displaystyle{\mathbb{P}}(D^{\epsilon}_{n}\cap F_{n}(k))$
$\displaystyle\leq{\mathbb{P}}\bigl{(}H^{(L)}(n,\delta,\epsilon)\bigr{)}+{\mathbb{P}}\bigl{(}H^{(R)}(n,\delta,\epsilon)\bigr{)}$
$\displaystyle\leq\sum_{l=-\lfloor 2m\sqrt{n}\sigma\rfloor}^{\lfloor
2m\sqrt{n}\sigma\rfloor}{\mathbb{P}}\bigl{(}\cup_{k\in{\mathbb{Z}}}H_{k}^{(L)}(n,\delta,\epsilon)\bigr{)}+\sum_{r=-\lfloor
2m\sqrt{n}\sigma_{0}\rfloor}^{\lfloor
2m\sqrt{n}\sigma\rfloor}{\mathbb{P}}\bigl{(}\cup_{k\in{\mathbb{Z}}}H_{r,k}^{(R)}(n,\delta,\epsilon)\bigr{)}$
$\displaystyle\leq 16m\sqrt{n}\sigma C_{3}(\delta)\epsilon C_{2}/\sqrt{\lfloor
n\gamma\delta\rfloor}\leq C_{1}(\delta,m)\epsilon$

for a proper choice of $C_{1}(\delta,m)$. This completes the proof. ∎

##  8 Appendix

In this section we present proofs of some basic results that we have used
before.

Proof of Lemma 4.4: Fix $j\geq 1$. Given
$h^{\sigma_{j}}({\mathbf{x}}_{1})={\mathbf{v}}_{j}$, we recall that the path
$\\{h^{n}({\mathbf{v}}_{j}):n\geq 1\\}$ uses the random vectors
$\\{\Lambda_{\mathbf{w}}:{\mathbf{w}}\in\mathbb{H}^{+}({\mathbf{v}}_{j}(2))\\}$
only and always stays within the region $\nabla({\mathbf{v}}_{j})$.

The main idea is to show that distribution of the point set
$V^{+}_{{\mathbf{v}}_{j}(2)}$ remains invariant with respect to reflection
about the line $x={\mathbf{v}}_{j}(1)$. For
${\mathbf{w}}={\mathbf{v}}_{j}+(s,t)\in\mathbb{H}^{+}({\mathbf{v}}_{j}(2))$ we
define its reflected copy reflected about the line $X={\mathbf{v}}_{j}(1)$ as

$\overline{{\mathbf{w}}}:={\mathbf{v}}_{j}+(-s,t).$

Now, using the collection
$\\{\Gamma_{\mathbf{w}}=(B_{\mathbf{w}},R_{\mathbf{w}},\Lambda_{\mathbf{w}}):{\mathbf{w}}\in{\mathbb{Z}}^{2}\\}$,
we define a new collection
$\\{\Gamma^{\prime}_{\mathbf{w}}:=B_{{\mathbf{w}}}^{\prime},R_{{\mathbf{w}}}^{\prime},\Lambda^{\prime}_{\mathbf{w}}:{\mathbf{w}}\in{\mathbb{Z}}^{2}\\}$
given by

$\displaystyle\bigl{(}B_{{\mathbf{w}}}^{\prime},R_{{\mathbf{w}}}^{\prime},\Lambda^{\prime}_{\mathbf{v}}:=(X^{\prime}_{{\mathbf{w}}},Y^{\prime}_{{\mathbf{w}}})\bigr{)}:=\begin{cases}\bigl{(}B_{\overline{{\mathbf{w}}}},-R_{\overline{{\mathbf{w}}}},(-X_{\overline{{\mathbf{w}}}},Y_{\overline{{\mathbf{w}}}})\bigr{)}&\text{
if }{\mathbf{w}}\in\mathbb{H}^{+}({\mathbf{v}}_{j}(2))\\\
\bigl{(}B_{{\mathbf{w}}},R_{{\mathbf{w}}},(X_{{\mathbf{w}}},Y_{{\mathbf{w}}})\bigr{)}&\text{
if }{\mathbf{w}}\in\mathbb{H}^{-}({\mathbf{v}}_{j}(2)).\end{cases}$

The resultant point process generated from this collection is defined as

$V^{\prime}:=\\{{\mathbf{w}}+\Lambda^{\prime}_{\mathbf{w}}:B_{\mathbf{w}}^{\prime}=1\\}.$

We observe that the newly created point process $V^{\prime}$ on
$\mathbb{H}^{+}({\mathbf{v}}_{j}(2))$ gives a reflected copy of the set
$V^{+}_{{\mathbf{v}}_{j}(2)}$ reflected about the line
$x={\mathbf{v}}_{j}(1)$. More precisely, a point ${\mathbf{w}}$ belongs to
$(V^{\prime})^{+}_{{\mathbf{v}}_{j}(2)}$ if and only if
$\overline{{\mathbf{w}}}\in V^{+}_{{\mathbf{v}}_{j}(2)}$.

It is also important to observe that for
${\mathbf{w}}\in\mathbb{H}^{+}({\mathbf{v}}_{j}(2))$ we have
$R_{{\mathbf{w}}}^{\prime}=-R_{\overline{{\mathbf{w}}}}$. This ensures that in
case of ‘tie’ (w.r.t. nearest member from $V^{\prime}$ at the next level) the
choice of outgoing edge gets reversed appropriately. This way the newly
constructed collection
$\\{\Gamma^{\prime}_{\mathbf{w}}:{\mathbf{w}}\in{\mathbb{Z}}^{2}\\}$ ensures
that

$\displaystyle\overline{h^{n}}({\mathbf{v}}_{j},V)=\overline{h^{n}}({\mathbf{v}}_{j},V^{+}_{{\mathbf{v}}_{j}(2)})=h^{n}({\mathbf{v}}_{j},(V^{\prime})^{+}_{{\mathbf{v}}_{j}(2)})\text{
for all }n\geq 1.$ (64)

The first equality follows as ${\mathbf{v}}_{j}$ is a renewal step. As the
region $\nabla({\mathbf{v}}_{j})$ is symmetric about the line
$X={\mathbf{v}}_{j}(1)$, (64) ensures that the new point process satisfies the
renewal conditions as well:

  * (i)

$h^{n}({\mathbf{v}}_{j},V^{\prime})=h^{n}({\mathbf{v}}_{j},(V^{\prime})^{+}_{{\mathbf{v}}_{j}(2)})\in\nabla({\mathbf{v}}_{j})$
for all $n\geq 1$.

  * (ii)

The event $\text{Out}({\mathbf{v}}_{j})$ occurs w.r.t. the point set
$V^{\prime}$ also as
$(V^{\prime})^{+}_{{\mathbf{v}}_{j}(2)}=V^{-}_{{\mathbf{v}}_{j}(2)}$.

Hence, given $h^{\sigma_{j}}({\mathbf{x}}_{1})={\mathbf{v}}_{j}$, distribution
of the process $\\{h^{n}({\mathbf{v}}_{j}):n\geq 1\\}$ starting from
${\mathbf{v}}_{j}$ remains the same when it progresses using the collection
$\\{\Lambda^{\prime}_{\mathbf{v}}:{\mathbf{v}}\in\mathbb{H}^{+}({\mathbf{v}}_{j}(2))\\}$.

Let ${\mathbf{v}}_{j+1}={\mathbf{v}}_{j}+(s_{0},t_{0})$ be the point of the
next renewal w.r.t. the point process $V$. It suffices to show that the point
$\overline{{\mathbf{v}}}_{j+1}$ is the position of the next renewal with
respect to the point set $V^{\prime}$ as well. Now, (64) ensures that we have
$h^{t_{0}}({\mathbf{v}}_{j},V^{\prime})=\overline{{\mathbf{v}}}_{j+1}$. From
(64) we also obtain that for any $1\leq n_{1}\leq n_{2}$

$\displaystyle\bigl{(}h^{n_{2}}({\mathbf{v}}_{j},V)-h^{n_{1}}({\mathbf{v}}_{j},V)\bigr{)}(1)=-\bigl{(}h^{n_{2}}({\mathbf{v}}_{j},V^{\prime})-h^{n_{1}}({\mathbf{v}}_{j},V^{\prime})\bigr{)}(1).$
(65)

By taking $n_{1}=t_{0}$ and $n_{2}\geq t_{0}+1$ in Equation (65) we obtain

$h^{n}(\overline{{\mathbf{v}}}_{j+1},V^{\prime})=h^{n}(\overline{{\mathbf{v}}}_{j+1},(V^{\prime})^{+}_{{\mathbf{v}}_{j+1}(2)})\in\nabla(\overline{{\mathbf{v}}}_{j+1})\text{
for all }n\geq 1.$

This implies occurrence of the event
$\text{In}^{+}(\overline{{\mathbf{v}}}_{j+1})$ w.r.t. the point process
$V^{\prime}$. Given that the event $\text{Out}({\mathbf{v}}_{j})$ has
occurred, occurrence of the event $\text{Out}(\overline{{\mathbf{v}}}_{j+1})$
w.r.t. the point set $V^{\prime}$ depends only on the random vectors
$\\{\Lambda^{\prime}_{\mathbf{w}}:{\mathbf{w}}(2)\in[{\mathbf{v}}_{j}(2)+1,{\mathbf{v}}_{j+1}(2)]\\}$.
We show that for ${\mathbf{w}}\in{\mathbb{Z}}^{2}$ with
${\mathbf{w}}(2)\in[{\mathbf{v}}_{j}(2)+1,{\mathbf{v}}_{j+1}(2)]$, we have

${\mathbf{w}}+(X_{\mathbf{w}},Y_{\mathbf{w}})\in\nabla({\mathbf{v}}_{j+1})\text{
if and only if
}\overline{{\mathbf{w}}}+(-X_{\overline{{\mathbf{w}}}},Y_{\overline{{\mathbf{w}}}})\in\nabla(\overline{{\mathbf{v}}}_{j+1}).$

This follows from the fact that

$\displaystyle({\mathbf{w}}-{\mathbf{v}}_{j+1})(1)+X_{\mathbf{w}}$
$\displaystyle=-(\overline{{\mathbf{w}}}-\overline{{\mathbf{v}}}_{j+1})(1)-X_{\overline{{\mathbf{w}}}}\text{
and }$ $\displaystyle({\mathbf{w}}-{\mathbf{v}}_{j+1})(2)+Y_{\mathbf{w}}$
$\displaystyle=(\overline{{\mathbf{w}}}-\overline{{\mathbf{v}}}_{j+1})(2)+Y_{\overline{{\mathbf{w}}}}.$

Hence, occurrence of the event $\text{Out}({\mathbf{v}}_{j+1})$ w.r.t. the
point process $V$ implies and implied by occurrence of the event
$\text{Out}(\overline{{\mathbf{v}}}_{j+1})$ w.r.t. the point process
$V^{\prime}$.

Essentially, we proved that for any $n\geq 1$, the $n$-th step
$h^{n}({\mathbf{v}}_{j})={\mathbf{x}}$ is a renewal step (w.r.t. the point
process $V$) if and only if the corresponding step w.r.t. $V^{\prime}$ is
given by $h^{n}({\mathbf{v}}_{j},V^{\prime})=\overline{{\mathbf{x}}}$ is a
renewal step as well. This proves that starting from ${\mathbf{v}}_{j}$ the
step

$h^{t_{0}}({\mathbf{v}}_{j},V^{\prime})=\overline{{\mathbf{v}}}_{j+1}$

gives the next renewal step w.r.t. $V^{\prime}$. This completes the proof. ∎

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