Quenched and averaged tails of the heat kernel of the two-dimensional uniform spanning tree

This paper is dedicated to the memory of Harry Kesten, a pioneer in the study of anomalous random walks in random media

Abstract

This article investigates the heat kernel of the two-dimensional uniform spanning tree. We improve previous work by demonstrating the occurrence of log-logarithmic fluctuations around the leading order polynomial behaviour for the on-diagonal part of the quenched heat kernel. In addition we give two-sided estimates for the averaged heat kernel, and we show that the exponents that appear in the off-diagonal parts of the quenched and averaged versions of the heat kernel differ. Finally, we derive various scaling limits for the heat kernel, the implications of which include enabling us to sharpen the known asymptotics regarding the on-diagonal part of the averaged heat kernel and the expected distance travelled by the associated simple random walk.

Appendix: Short LERW paths

In this section we improve the estimates in [8] to prove Theorem 2.7. We begin by considering the following situation, which is described in terms of parameters $m,n,N\in {\mathbb {N}}$ satisfying $4\le n \le m \le m+2n \le N$, cf. [8, Definition 1.4]. Let $B_m = {B_\infty }(0,m)$, $B_N={B_\infty }(0,N)$, and $x \in \partial _R B_m$, where for a square B we write $\partial _R B$ for the right-hand side of the interior boundary of B. Moreover, let $x_1= x + (\frac{n}{2},0)$, and define $A_n(x) = {B_\infty }(x,n/4)$. Finally, we also suppose we are given a subset $K \subseteq B_m$ that contains a path in $B_m$ from 0 to x. Importantly, we note that the latter assumption was not made in [8]; it is the key to removing the terms in $\log (N/n)$ in [8, Lemmas 4.6 and 6.1, and Propositions 6.2 and 6.3]. We also remark that in [8] the balls $B_n$ and $B_N$ were in the $\ell _2$ norm on ${{\mathbb {Z}}}^2$ rather than the $\ell _\infty$ norm, but this makes no essential difference to the arguments.

The first result of the section concerns the Green’s function G of a simple random walk S on ${\mathbb {Z}}^2$. Given a subset $A\subsetneq {\mathbb {Z}}^2$, we write $G_A(y,z)$ for the expected number of visits that S makes to z when it starts at y up until it exits A. In the proof, we write ${{\mathbf {P}}}_x$ for the law of the random walk started from x, and ${{\mathbf {E}}}_x$ for the corresponding expectation.

Lemma A.1

There exist constants $c_i$ such that, for $y,z \in A_n(x)$,

$$\begin{aligned} c_1 \log \left( \frac{n}{1\vee |y-z|}\right) \le G_{B_N \backslash K }(y,z) \le c_2 \log \left( \frac{n}{1\vee |y-z|}\right) . \end{aligned}$$

(A.1)

Proof

Set $A_1= B_{5n/16}(x_1)$ and $A_2= B_{3n/8}(x_1)$. We note that

$$\begin{aligned} c_1 \log \left( \frac{n}{1\vee |y-z|}\right) \le G_{A_1}(y,z) \le G_{A_2}(y,z)\le c_2 \log \left( \frac{n}{1\vee |y-z|}\right) . \end{aligned}$$

(Cf. The applications of results from [27, Chapter 6] that appear as [8, Proposition 2.4].) Hence, since $G_{B_N\backslash K }(y,z) \ge G_{A_1}(y,z)$, the lower bound is immediate. For the upper bound, writing $T_A$ and $\tau _A$ for the hitting and exit time of a subset $A\subseteq {\mathbb {Z}}^2$ by the simple random walk S, respectively, we have

$$\begin{aligned} G_{B_N \backslash K }(y,z)&= G_{A_2}(y,z) + {{\mathbf {E}}}_y\left( G_{B_N \backslash K }( S_{\tau _{A_2}},z)\right) \\&\le c_2 \log \left( \frac{n}{1\vee |y-z|}\right) + \max _{w \in \partial A_2} {{\mathbf {P}}}_w( T_{A_1} < T_K) \max _{w' \in \partial A_1} G_{B_N \backslash K }(w',z). \end{aligned}$$

By the discrete Harnack inequality (see [27, Theorem 6.3.9], for example) and the fact that K contains a path from x to 0, we have that ${{\mathbf {P}}}_w( T_{A_1} < T_K) \le 1-c_3$ for all $w \in \partial A_2$. Further, for $w' \in \partial A_1$ we have

$$ {{\mathbf {P}}}_{w'}( T_z < \tau _{A_2}) \le 1\wedge \frac{c_4}{\log n} . $$

(Again, cf. [8, Proposition 2.4].) Combining these estimates gives $G_{B_N \backslash K }(z,z) \le c_2 \log (n) +(1-c_3)G_{B_N \backslash K }(z,z)$, and thus $G_{B_N \backslash K }(z,z)\le \frac{c_2}{c_3}\log (n)$. Hence

$$\begin{aligned} G_{B_N \backslash K }(y,z)&\le c_2 \log \left( \frac{n}{1\vee |y-z|}\right) +c_5, \end{aligned}$$

which yields the bound (A.1). $\square$

Next, let ${\tilde{S}}$ be a random walk started at x and conditioned to leave $B_N$ before its first return to K. We write ${\tilde{G}}(\cdot , \cdot )$ for the Green’s function of ${\tilde{S}}$.

Lemma A.2

(Cf. [8, Lemma 4.6]). There exist constants $c_i$ such that, for $z \in A_n(x)$ we have $c_1 \le {\tilde{G}}(x,z) \le c_2$.

Proof

We follow the proof in [8]. Taking $y=z$ in (A.1) we can improve the upper bound on $G_{B_N \backslash K }(z,z)$ in [8, (4.10)] to $c \log n$. Using Lemma A.1 again, we can improve the upper bound in the equation above [8, (4.11)], and hence improve the upper bound in [8, (4.11)] from $c \log (N/n)/\log N$ to $c /\log n$. With these new bounds the argument of [8, Lemma 4.6] gives that ${\tilde{G}}(x,z) \le c_2$. $\square$

The following two results refine some conditional hitting time estimates from [8].

Lemma A.3

(cf. [8, (6.1)]). There exists a constant $c_1$ such that if $D_1 = \partial _R {B_\infty }(x,n/16)$ and $K'=K\backslash \{x\}$, then, for $v \in D_1$,

$$\begin{aligned} {{\mathbf {P}}}_v \left( T_x< \tau _{{B_\infty }(x,n/8)} \,|\, T_x <T_{K'} \wedge \tau _{B_N} \right) \ge c_1>0. \end{aligned}$$

Proof

Write $B'={B_{n/8}(x)}$. The second displayed equation on [8, p. 2409] gives

$$\begin{aligned}&{{\mathbf {P}}}_v \left( T_x< \tau _{B_{n/8}(x)} \,|\, T_x<T_{K'} \wedge \tau _{B_N} \right) \nonumber \\&\quad = \frac{ G_{B'\backslash K}(v,v) }{ G_{B_N\backslash K}(v,v) }\times \frac{ {{\mathbf {P}}}_x ( T_v< \tau _{B'} \wedge T^+_K ) }{ {{\mathbf {P}}}_x ( T_v < \tau _{B_N} \wedge T^+_K )}, \end{aligned}$$

(A.2)

where $T^+_K= \min \{ j \ge 1: S_j \in K\}$. As in Lemma A.1 we have that $G_{B_N\backslash K}(v,v) \le c \log n$, and so the ratio of Green’s functions in (A.2) is bounded below by a constant $c>0$. Using the strong Markov property at $\tau _{B'}$ we obtain

$$\begin{aligned} {{\mathbf {P}}}_x ( T_v< \tau _{B_N} \wedge T^+_K )&\le {{\mathbf {P}}}_x ( T_v< \tau _{B'} \wedge T^+_K ) \\&\quad + {{\mathbf {P}}}_x ( \tau _{B'} \le T^+_K ) \max _{y \in \partial B'} {{\mathbf {P}}}_y( T_v < \tau _{B_N} \wedge T^+_K ). \end{aligned}$$

The argument at the top of [8, p. 2410] gives that

$$ {{\mathbf {P}}}_x ( \tau _{B'} \le T^+_K ) \le c (\log n) \, {{\mathbf {P}}}_x ( T_v < \tau _{B'} \wedge T^+_K ) . $$

Moreover, for $y \in \partial B'$,

$$ {{\mathbf {P}}}_y(T_v < \tau _{B_N} \wedge T^+_K ) \le \frac{ G_{{{\mathbb {Z}}}^2 \backslash K}(y,v) }{ G_{{{\mathbb {Z}}}^2\backslash K}(v,v) }, $$

and as in Lemma A.1 we have $G_{{{\mathbb {Z}}}^2 \backslash K}(y,v) \le c$, $G_{{{\mathbb {Z}}}^2 \backslash K}(v,v) \ge c \log n$. Combining these estimates concludes the proof. $\square$

Lemma A.4

(cf. [8, (6.2)]). There exists a constant $c>0$ such that if $w \in \partial _R {B_\infty }(x,n)$, then

$$\begin{aligned} {{\mathbf {P}}}_w \left( \tau _{B_N}< T_{{B_\infty }(x,7n/8)} \,|\, \tau _{B_N} < T_K \right) \ge c. \end{aligned}$$

(A.3)

Proof

As on [8, p. 2410], we let $y_0$ be the point in $B_n(x)$ that maximises ${{\mathbf {P}}}_y( \tau _{B_N} < T_K )$. Writing $B_7= {B_\infty }(x,7n/8)$, $T_7= T_{B_7}$, we have

$$\begin{aligned} {{\mathbf {P}}}_{y_0} ( \tau _{B_N}< T_K )&= {{\mathbf {P}}}_{y_0}( \tau _{B_N}< T_K \wedge T_7 ) + {{\mathbf {E}}}^{y_0} ( {\mathbf {1}}_{\{T_7< \tau _{B_N} \wedge T_K\} } {{\mathbf {P}}}_{S_{T_7}}( \tau _{B_N}< T_K ) ) \\&\le {{\mathbf {P}}}_{y_0}( \tau _{B_N}< T_K \wedge T_7 ) + \max _{v \in \partial B_7} {{\mathbf {P}}}_v( \tau _{B_N} < T_K). \end{aligned}$$

Since K contains a path from 0 to x, the discrete Harnack inequality (again, see [27, Theorem 6.3.9], for example) gives us that there exists a constant $p_1>0$ such that

$$ {{\mathbf {P}}}_v( \tau _{{B_\infty }(x,n)} < T_K) \le 1-p_1,\qquad \text{ for } \text{ all } v \in \partial B_7.$$

Thus

$$ {{\mathbf {P}}}_{y_0} ( \tau _{B_N}< T_K ) \le {{\mathbf {P}}}_{y_0}( \tau _{B_N}< T_K \wedge T_7 ) + (1-p_1) {{\mathbf {P}}}_{y_0} ( \tau _{B_N} < T_K ), $$

which proves (A.3) in the case $w=y_0$. We can now use a reflection argument as on [8, p. 2410-2411] to obtain the general case. $\square$

These estimates now lead to an improved lower bound on the length of a LERW. Recall the definition of the conditioned r.w. ${{\tilde{S}}}$, and set $L_1 = {{\mathcal {L}}}( {{\mathcal {E}}}_{B_N}({\tilde{S}}))$, $L_2 = {{\mathcal {E}}}_{B_n(x)}(L_1)$.

Lemma A.5

(cf. [8, Lemma 6.1]). There exists a constant $c>0$ such that, for any $z \in A_n(x)$,

$$\begin{aligned} {{\mathbf {P}}}( z \in L_2 ) \ge c n^{\kappa -2}. \end{aligned}$$

Proof

Using Lemmas A.4 and A.3 to replace [8, (6.1),(6.2)], this follows as in [8]. $\square$

Proposition A.6

(cf. [8, Proposition 6.2 and 6.3]). There exist constants $c_1,c_2$ and $p>0$ such that

$$\begin{aligned}&c_1 n^\kappa \le {{\mathbf {E}}}M \le c_2 n^\kappa , \nonumber \\&{{\mathbf {E}}}(M^2 ) \le c_2 n^{2\kappa }, \nonumber \\&{{\mathbf {P}}}( M \le c_3 n^\kappa ) \le 1-p. \end{aligned}$$

(A.4)

Proof

Given Lemmas A.5 and A.2 the bounds on ${{\mathbf {E}}}(M)$ and ${{\mathbf {E}}}(M^2)$ follow as in [8]. The final inequality is then immediate from a second moment bound. $\square$

Proof of Theorem 2.7

We follow the proof of [8, Proposition 6.6], first proving the result in the case when $D= B_N(0)$, where $N/2 \le nk \le 3N/4$ for some $k \ge 4$. Set $L = {{\mathcal {L}}}( {{\mathcal {E}}}_{B_N(0)}(S^0))$, and, for $j=1, \dots k$, let $\gamma _j = {{\mathcal {E}}}_{B_{j n}(0)}(L)$. Let $x_j$ be the point where L first exits $B_{j n}(0)$, and $B_j = B_{n}(x_j)$. Let $\alpha _j$ be the path L from $x_{j-1}$ to its first exit from $B_{j-1}$, and $V_j$ be the number of hits by $\alpha _j$ on the set $B_{j-1}$. Let ${{\mathcal {F}}}_j$ be the $\sigma$-field generated by $\gamma _j$. Using the domain Markov property for the LERW (see [23]) and then (A.4), we have

$$\begin{aligned} {{\mathbf {P}}}\left( V_j \le c_3 n^\kappa | {{\mathcal {F}}}_{j-1} \right) = {{\mathbf {P}}}\left( M^{\gamma _{j-1}}_{(j-1)n,n,N,x_{j-1}} \le c_3 n^\kappa \right) \le 1-p. \end{aligned}$$

(A.5)

Let $\eta _j = {\mathbf {1}}_{\{V_j \le c_3 n^\kappa \}}$. By (A.5), $\sum _{j=1}^k \eta _j$ stochastically dominates a binomial random variable with parameters k and p, and so there exists a constant $c>0$ such that

$$ {{\mathbf {P}}}\left( \sum _{j=1}^k \eta _j < {\tfrac{1}{2}}p k \right) \le e^{-c k}. $$

Setting $L' = {{\mathcal {E}}}_{B_{nk}(0)}(L)$, we have $| L' | \ge c_3 n^\kappa \sum _{j=1}^k \eta _j$, and thus as $N/2 \le nk \le 3N/4$ we obtain

$$\begin{aligned} {{\mathbf {P}}}\left( |L' | < c k^{-1/4} N^\kappa \right) \le e^{-c k}; \end{aligned}$$

taking $k = c \lambda ^{1/(\kappa -1)} = c\lambda ^4$ this gives the result when $D=B_N$. Note that the proof above actually gives the lower bound for the length of $L'$ rather than L, so we can use Lemma 2.1 with $D_1= B_N$, $D_2=D$ to obtain a lower bound of the same form for $|{{\mathcal {E}}}_{B_N(0)} ({{\mathcal {L}}}({{\mathcal {E}}}_{D}(S^0)))|$. $\square$