Extremes of local times for simple random walks on symmetric trees

We consider local times of the simple random walk on the $b$-ary tree of depth $n$ and study a point process which encodes the location of the vertex with the maximal local time and the properly centered maximum over leaves of each subtree of depth $r_n$ rooted at the $(n-r_n)$ level, where $(r_n)_{n \geq 1}$ satisfies $\lim_{n \to \infty} r_n = \infty$ and $\lim_{n \to \infty} r_n/n \in [0, 1)$. We show that the point process weakly converges to a Cox process with intensity measure $\alpha Z_{\infty} (dx) \otimes e^{-2\sqrt{\log b}~y}dy$, where $\alpha>0$ is a constant and $Z_{\infty}$ is a random measure on $[0, 1]$ which has the same law as the limit of a critical random multiplicative cascade measure up to a scale factor. As a corollary, we establish convergence in law of the maximum of local times over leaves to a randomly shifted Gumbel distribution.


Introduction
Much efforts have been made in the study of the so-called log-correlated random field such as the branching Brownian motion (BBM), the branching random walk (BRW), and the two-dimensional discrete Gaussian free field (DGFF). One of the remarkable features of these models is that laws of their maxima share common properties: each of the laws weakly converges to a randomly shifted Gumbel distribution [33,1,20]. It is believed that each of the limiting extremal processes of a wide class of log-correlated fields converges to a so-called randomly shifted decorated Poisson point process [38] and it is established for the BBM [2,5], the BRW [36], and the two-dimensional DGFF [14].
It is well-known that local times of random walks on graphs have close relationships with DGFFs thanks to "the generalized second Ray-Knight theorem" [29] (this goes back to the Dynkin isomorphism [28]) which has many applications, for example, to the cover time [24,23,39]. Since the occupation time field of the simple random walk on the tree or on the two-dimensional lattice is closely related to the BRW or two-dimensional DGFF respectively, it is natural to expect that their maxima and cover times belong to the universal class mentioned above: it is known that the cover times have subleading terms similarly to other log-correlated fields [26,9] and that the cover time of the binary tree is tight [19,10], but further details are still open.
In this paper, we consider local times of the simple random walk on the b-ary tree of depth n at time much larger than the maximal hitting time and study convergence of a point process encoding extreme local maxima of the local times as n → ∞.
To state our result, we begin with some notation. We fix an arbitrary integer b ≥ 2 throughout the paper. We will write T to denote the b-ary tree with root ρ: this is a rooted tree whose vertices have exactly b children. Let T i be the ith generation of T . Set T ≤n := ∪ n i=0 T i . For v ∈ T , we will write |v| to denote the depth of v. For u ∈ T , let T u be the subtree of T rooted at u, and we define T u i and T u ≤n similarly. For v, u ∈ T , let v ∧ u be the most recent common ancestor of v and u. Let X = (X t , t ≥ 0, P v , v ∈ T ≤n ) be the continuous-time simple random walk on T ≤n with exponential holding times of parameter 1. We define the local time of X by where deg(v) is the degree of v, and the inverse local time by τ(t) := inf{s ≥ 0 : L n s (ρ) > t}, t ≥ 0.
Let E(T ) be the set of all edges on T . Let (Y e ) e∈E(T ) be independent and identically distributed random variables whose common law is the normal distribution with mean 0 and variance 1/2. To each v ∈ T , we assign h v := ∑ |v| i=1 Y e v i , where e v 1 , . . . , e v |v| are the edges on the unique shortest path from ρ to v. We will call (h v ) v∈T a BRW on T . It is well-known that the so-called derivative martingale For each n ∈ N and x ∈ [0, 1], let v(x) be the vertex in T n with x ∈ [σ (v(x)), σ (v(x)) + b −n ]. (If we have two such vertices, we choose the one whose location is the largest.) We define the random measure called a (critical) random multiplicative cascade measure by Z n (dx) where dx is the Lebesgue measure on [0, 1]. Barral, Rhodes, and Vargas [7] observed that the weak limit Z ∞ := lim n→∞ Z n exists almost surely. (1.4) For each v ∈ T n , set The random measure Z ∞ satisfies that where " d =" means that the laws of the left and the right are the same and D (v) ∞ , v ∈ T n are independent copies of D ∞ which are independent of (h v ) v∈T n . See [8,22] for more details on Z ∞ . For each (x, y) ∈ [0, 1] × R, we write δ (x,y) to denote the Dirac measure at (x, y). For each 0 ≤ m ≤ n, we define the point process on , (1.5) where the centering sequence a n (t) is given by and for each u ∈ T n−m , argmax u L n τ(t) is the vertex v * on T u m ⊂ T n with L n τ(t) (v * ) = max v∈T u m L n τ(t) (v). (If two or more vertices on T u m attain the maximum, we take the one whose location is the largest among such vertices.) We regard Ξ Since this space is metrizable as a complete separable metric space, we can consider convergence in law of sequences of random measures. Given a random measure ν on [0, 1] × R, we will write PPP(ν) to denote a point process on [0, 1] × R which, conditioned on ν, is a Poisson point process with intensity measure ν (that is PPP(ν) is a Cox process). We now state the main result of this paper: Theorem 1.1 There exists c 1 > 0 such that for any sequence (t n ) n≥1 with lim n→∞ √ t n n = θ ∈ [0, ∞] and t n ≥ c 1 n log n for each n ∈ N, and any sequence (r n ) n≥1 with lim n→∞ r n = ∞ and lim n→∞ r n /n ∈ [0, 1), the point process Ξ (r n ) n,t n converges in law to a Cox process as n → ∞, where Z ∞ is the random measure on [0, 1] in (1.4),  [4] and the two-dimensional DGFF [12,13]. Our setting is inspired by [15]. The convergence of the full extremal process has been established for the BBM [2,5], the BRW [36], and the two-dimensional DGFF [14]. Related convergence for the local times on the b-ary tree will be studied in a sequel paper.
By Theorem 1.1 and a tail estimate of the maximum of local times over leaves (Proposition 3.1(i) below), we have: There exists c 1 > 0 such that for all λ ∈ R and any sequence (t n ) n≥1 with lim n→∞ √ t n n = θ ∈ [0, ∞] and t n ≥ c 1 n log n for each n ∈ N, where D ∞ , β * and γ * are given by (1.1), (1.8) and (1.9), respectively.

Remark 1.4 Let (h v ) v∈T be a BRW on T . By Theorem 2.2 and Lemma A.7, one can
show that for all λ ∈ R and any sequence (t n ) n≥1 with lim n→∞ √ t n /n 2 = ∞, where γ * is given in (1.9) and the centering sequence m n is defined by (Note that the convergence of the maximum of the BRW has already been established in [6,1,21].) The centering sequence a n (t n ) in (1.10) is different from m n by the term The organization of the paper is as follows. Section 2 gives preliminary lemmas which we use repeatedly throughout the paper. In Section 3, we obtain tail probabilities of the maximum of local times over leaves which are essential to next sections. One can find that for each leaf v, the law of the local time process along the path from ρ to v is the same as that of a zero-dimensional squared Bessel process (see Lemma 2.3). By this and the Markov property of local time processes (see Lemma 2.1), roughly speaking, one can regard the field of local times over the set of leaves as a branching Bessel process. This gives hints of how to estimate the tail of the maximum of local times over leaves: we use the constraint first and second moment methods developed in the BBM, BRW, and two-dimensional DGFF settings. (See, for example, [17,1,20]. We especially use techniques in [27,20].) Typical behavior of a vertex with extreme local time is as follows: the local time process along the path from the root to the vertex stays below a curve and finally reach the maximal value at the vertex (see the proof of Proposition 3.1). In Section 4, we show that two leaves with local times near maxima are either very close or far away. This suggests that local maximizers are distributed as a Poisson nature. More technically, this implies that Ξ with probability tending to 1 as n → ∞ and then q → ∞, which is one of the key steps in the proof of Theorem 1.1. In Section 5, we obtain a limiting tail of the maximum of local times over leaves which is crucial to study the Laplace functional of Ξ (n−q) n,t n . In the estimate, entropic repulsion (Lemma 2.4(ii)) plays an important role: this enables us to compute the tail of the maximum by using the reflection principle of a Brownian motion. In Section 6, we give the proof of Theorem 1.1 and Corollary 1.3.
We should emphasize that it is more convenient to study "continuous" version of local times rather than the original "discrete" ones especially when we estimate tail probabilities of the maximum of the local times over leaves. To take the advantage, motivated by [34,39], we consider the local time process of the Brownian motion on the associated metric tree as the "continuous" version.
We will write c 1 , c 2 , . . . to denote positive universal constants whose values are fixed within each argument. We use c 1 (M), c 2 (M) . . . for positive constants which depend on M. Given sequences (c n ) n≥1 and (c ′ n ) n≥1 , we write c ′ n = O(c n ) if there exists a universal constant C such that |c ′ n /c n | ≤ C for all n ≥ 1. We write |S| to denote the cardinality of a set S.

Preliminary lemmas
In this section, we collect some lemmas which we use repeatedly throughout the paper. We first recall the metric tree and the Brownian motions on it. In the study of local times of random walks on graphs, Lupu [34] and Zhai [39] used the corresponding metric graphs and Brownian motions. We follow the approach and find its advantages in obtaining precise tail probabilities of the maximum of local times over leaves on the b-ary tree. Given a graph G, we will write E(G) to denote the edge set of G. Let T be the b-ary tree. We regard each e ∈ E(T ) as an interval of length 1/2 by setting I e := {e} × 0, 1 2 . SetĪ e := I e ∪ {e − , e + }, where e − , e + ∈ T be the endpoints of the edge e. Let π e be the map from I e to 0, 1 2 defined by π e ((e, x)) := x. We extend π e to the map fromĪ e to 0, 1 2 by setting π e (e − ) := 0, π e (e + ) := 1 2 . We define a metric tree of depth n by For each k ∈ N and v ∈ T , we will write T v ≤k to denote the metric tree corresponding to the subtree T v ≤k . We define the metric d(·, ·) on T ≤n as follows: for x, y ∈ T ≤n , let e x and e y be the edges with x ∈ I e x and y ∈ I e y , respectively. In the case I e x = I e y , we define where d g is the graph distance on T ≤n . In the case I e x = I e y , we set d(x, y) := |π e x (x) − π e y (y)|. We define a measure m on T ≤n by where ν e := ν • π e , and ν is the Lebesgue measure on (0, 1/2). We have a m-symmetric Hunt process on T ≤n with continuous sample paths such that on each I e , it behaves like a standard Brownian motion on (0, 1/2) until it hits {e − , e + }, and when it starts at a vertex v, it chooses one of the edges incident to v uniformly at random, and moves on it as described above. See, for example, [30,32,34] for the construction. We write X = ( X t ,t ≥ 0, P x , x ∈ T ≤n ) to denote the process and call it a Brownian motion on T ≤n . It is known that X restricted to T ≤n behaves like a simple random walk on T ≤n in the following sense: for all v ∈ T ≤n and 1 ≤ i ≤ deg(v), Section 2], X has a space-time continuous local time { L n t (x) : (t, x) ∈ [0, ∞) × T ≤n } and the following holds for each v ∈ T ≤n under P v : where Exp(m) is an exponential random variable with mean m. We define the inverse local time by τ(t) := inf{s ≥ 0 : L n s (ρ) > t}, t > 0. By (2.1) and (2.2), we have The following is the Markov property of local times of the Brownian motion on T ≤n . The discrete version can be found in [23, Lemma 2.6].
Lemma 2.1 Fix n ∈ N,t > 0, and a ∈ T ≤n \T n . Let F ↑ be the σ -field generated by   39]) For all t > 0 and n ∈ N, on the same probability space, one can construct a local time (L n τ(t) (v)) v∈T ≤n and two BRWs (h v ) v∈T ≤n , (h ′ v ) v∈T ≤n on T ≤n satisfying the following: The construction of the coupling in Theorem 2.2 can be found in the proof of Theorem 3.1 of [39]. (Note that Zhai constructed the coupling in a more general setting and that in the context of [39], the law of the DGFF on a b-ary tree is the same as that of our BRW scaled by √ 2.) Let C[0, ∞) be the space of real-valued continuous functions on [0, ∞) and B (C[0, ∞)) be the σ -field generated by cylinder sets in C[0, ∞). We have a nice connection between the local time and the 0-dimensional squared Bessel process.
Note that our setting is different from that of [9,Lemma 7.7]. Notwithstanding, given Lemma A.1, the proof of Lemma 2.3 is almost the same as that of [9, Lemma 7.7], so we omit the proof of Lemma 2.3. It is known that the laws of 0-dimensional and 1dimensional squared Bessel processes are related to each other by the Radon-Nikodym derivative dQ 0 for all t > 0 and x > 0, where H 0 := inf{t ≥ 0 : X t = 0} and F t is the σ -field generated by {X s : s ≤ t}. See, for example, [9, (7.31)]. The transition semigroup {Q 0 t : t ≥ 0} of a 0-dimensional squared Bessel process is given by where δ 0 is the Dirac measure at 0. Q t (x, ·) in (2.7) has the density where I 1 (·) is the modified Bessel function of the first kind We will use the following asymptotic behavior of I 1 (·): (i) There exists c 1 > 0 such that for any z > 1, s > 0, and (ii) There exists δ z with lim z→∞ δ z = 0 such that for all z > 1, x ≤ 0, and s ≥ x 2 + z 2 , µ * s,z (x) ≤ (1 + δ z )µ s,z (x).

Tail of maximum of local time over leaves
The aim of this section is to obtain the following tail estimates of the maximum of local times of the simple random walk on the b-ary tree over leaves. Recall the definition of a n (t) from (1.6).
Recall the probability measure Q d x defined in Lemma 2.3 and set where X is a coordinate process. Fix δ ∈ (0, 1). By Lemma 2.3, P ρ (G n y (t)) is bounded from above by Fix 0 ≤ j ≤ n − 1. We first estimate I (2) j . By the strong Markov property of a 0dimensional squared Bessel process and (2.7), we have By the definition of τ, we have Assume that τ ∈ ( j, j + 1). Recall the definition of I 1 from (2.9). If z ≤ M( j+1−τ) 2 X τ , then we have , then by (2.10) and the assumption that M is sufficiently large, we have (3.10) By (3.9) and (3.10), we have where we have used the inequality ( j 2 for all z ∈ [0, 2δ 2 (m y,t,n ( j)) 2 ]. By (3.8) and (3.11), we have (3.12) where in the last inequality, we have used the assumption that y > M and M is sufficiently large.
Next, we will estimate I j is equal to is a Brownian motion on R with variance 1/2. Since the law of a 1-dimensional squared Bessel process is the same as that of a square of a standard Brownian motion on R, (3.13) is bounded from above by where we have used the translation invariance and Markov property of B in the last inequality.
Let P B j be the probability measure defined by By the Girsanov theorem, under P B j , the process is a Brownian motion on R with variance 1/2 started at 0. By the change of measure (3.15), the right of (3.14) is bounded from above by To estimate the tail of max 0≤s≤1 B s in the first term of the right of (3.17), we use the following: Similarly, in the case j = 0, by (2.6) and (3.18), we have Thus, by (3.19), (3.20), and the condition κ We now prove Proposition 3.1(i).

Proof of Proposition 3.1(i).
Recall the definitions of the event G n y (t) and the function g y,t,n (·) from (3.3) and (3.5). In view of Lemma 3.2, it is natural to impose the restriction that local time processes stay below the curve s → g y,t,n (s): we have Fix any v ∈ T n . Recall the process B from (3.16). By Lemma 2.3, (2.6), and the change of measure (3.15) (3.23) By Lemma 2.4(i), the right of (3.23) is bounded from above by Thus, by (3.22), (3.24), and Lemma 3.2, we have (3.1).
Next, we prove Proposition 3.1(ii). Fix δ ∈ (0, 1). For v ∈ T n , set the event To obtain Proposition 3.1(ii), we will apply the second moment method to ∑ v∈T n 1 A n v (t) . We first need the following: Proof. Fix any t ≥ n. By Lemma 2.3 and (2.6), P ρ (A n v (t)) is bounded from below by where we set , B n ∈ [a n (t) + y, a n (t) + y + 1) , ∃s ∈ [0, n], B n ∈ [a n (t) + y, a n (t) + y + 1) .
We first obtain an upper bound of J 2 . By using the density where we have used the symmetry of B in the first equality. Next, we obtain a lower bound of J 1 . Recall the process B from (3.16). By the change of measure (3.15) with j = n, we have By the reflection principle (2.11), for all n ≥ n 0 (n 0 is sufficiently large) and y ∈ [0, 2 √ n], (3.30) is bounded from below by n for all z ∈ [1/2, 1]. Thus, by (3.27), (3.29), and (3.31), we have (3.26).

To obtain upper bounds of
, u, v ∈ T n , we need the following: There exists c 3 > 0 such that for all n ∈ N, t > 0, v ∈ T n , and y ≥ 0, Proof. We first prove (1). Recall the process B from (3.16). By Lemma 2.3, (2.6), and the change of measure (3.15), the left of (3.32) is bounded from above by (3.34) By the reflection principle (2.11), (3.34) is bounded from above by where we have used the inequality 1 − e −x ≤ x for each x ≥ 0. Thus, we have obtained (3.32). The inequality (3.33) immediately follows from (1) with s = t and ℓ = 0.
Proof of Proposition 3.1(ii). Fix any n ≥ n 0 , t ≥ n and y ∈ [0, 2 √ n], where we take n 0 ∈ N large enough. Set We have The rest of the proof focuses on obtaining an upper bound of By Lemma 2.1, we have where we have used the independence of C ↓ v (s) and C ↓ u (s) for each s ≥ 0. (The independence follows from that of two types of excursions of a Brownian motion around By almost the same argument as the proof of Lemma 3.4 (1), we have (3.41) By (3.40) and (3.41), the right of (3.39) is bounded from above by (3.42) By (3.42), we have In the case ℓ = 0, by Lemma 3.4 (2), we have where we have used the independence of A n v (t) and A n u (t) for each v, u ∈ T n with |v ∧ u| = 0 in the first equality. (The independence follows from that of two types of excursions of a Brownian motion on T ≤n around ρ restricted to

Geometry of near maxima
In this section, we will prove that two leaves with local times near maxima are either very close or far away. More specifically, the following is the aim of this section.
For n ∈ N, t > 0, and k ∈ Z, set Since the law of the simple random walk on T ≤n ′ watched only on T ≤n is the same as that of the simple random walk on T ≤n , we have |Γ n ′ ,n In the proof of Proposition 4.1, we will use the following repeatedly.
We estimate the second term on the right-hand side of (4.3). By Lemma 2.1, we have We omit the subscript u in L ↓ s (x) and τ ↓ to simplify the notation.
We estimate each probability on the right-hand side of (4.4). Fix S ⊂ T n with |S| ≥ K and u ∈ S. Note that under the event that Γ n+r,n k (t) = S, we have √ t + a n+r (t) + y ≤ L n+r τ(t) (u) + a r L n+r τ(t) (u) + k + λ + 1.
By this and Proposition A.3 for t ≥ t 0 , where t 0 is sufficiently large, we have By (4.3)-(4.5), we have (ii) The proof of (ii) is almost the same as that of (i), so we omit the detail.
For the rest of this section, we focus on proving the following.
There exist c 1 > 0, n 0 , s 0 ∈ N, and t 0 > 0 such that for all n ≥ n 0 , t ≥ t 0 , and s 0 ≤ s ≤ n − s 0 , P ρ ∃v, u ∈ T n with |v ∧ u| = s : Before we prove this, let us show that Lemma 4.5 implies Proposition 4.1.
Proof of Proposition 4.1 via Lemma 4.5. Fix any n ≥ n 0 , t ≥ t 0 , and r 0 ≤ r ≤ n/4, where we take n 0 , r 0 ∈ N and t 0 > 0 sufficiently large. By Lemma 4.5 with c = 5 and c = 11 16 √ log b , the left of (4.1) is bounded from above by This is bounded from above by c 2 r −1/8 .
Proof of Lemma 4.5. Fix any n ≥ n 0 , t ≥ t 0 , s 0 ≤ s ≤ n − s 0 , where we take n 0 , s 0 ∈ N and t 0 > 0 sufficiently large. Set z := c log(s ∧ (n − s)). The left of (4.7) is bounded from above by For k ∈ Z, we set j * (k, z) := ⌈max{|k|, z}⌉. Fix sufficiently large constant t * > 0. We will decompose (4.8) into three terms with respect to k and j: For each w ∈ T s , let L ↓ be a local time of a Brownian motion on T w ≤n−s and set τ ↓ (q) := inf{p ≥ 0 : L ↓ p (w) > q}. We omit the subscript w in L ↓ and τ ↓ (q).
where J k, j 1 (S, w, w 1 , w 2 ) is given by (4.11) Fix S ⊂ T s with |S| ∈ I k, j , w ∈ S, and w 1 , w 2 ∈ T w 1 with w 1 = w 2 . By Lemma 2.1 and the independence of two types of excursions restricted to T w 1 ≤n−s−1 ∪ I {w,w 1 } or to (4.12) By the symmetry of the b-ary tree and (2.3), (4.12) is bounded from above by By (4.13), (4.10) is bounded from above by (4.14) We estimate (4.14) in different ways according to three cases: (a) k ≥ −z and j ≥ 0; (b) k < −z; (c) j < 0. In the case (a), we use Proposition 3.1(i) , Lemma 4.4, and Remark 4.3. In the case (b), we only use Lemma 4.4 and Remark 4.3 and estimate the square of the probability in (4.14) just by 1. In the case (c), we only use Proposition 3.1(i) and estimate the probability in the first display of (4.14) just by 1. Note that for k ∈ Z with k ≥ −z, we have √ t + a n (t) − clog(s ∧ (n − s)) Recall that z = c log(s ∧ (n − s)). By these observations, for sufficiently large t * , the first term of (4.9) is bounded from above by Next, we estimate the second term of (4.9). By Lemma 4.4 and Remark 4.3, we have Finally, we estimate the third term of (4.9). Fix k where J k 3 (w, w 1 , w 2 ) is given by (4.18) By (4.18) together with the symmetry of the b-ary tree, (4.17) is bounded from above by Since t 0 and n 0 are sufficiently large, we have √ t + a n (t) − clog(s ∧ (n − s)) + z ≥ (t * + 1) + a n−s ((t * + 1) 2 ) By Proposition 3.1(i) and (4.19), the third term of (4.9) is bounded from above by

Limiting tail of the maximum of local times
The aim of this section is to prove the exact asymptotics of the tail of the maximum of local times. Recall the constants β * and γ * from (1.8) and (1.9).
where we have used the inequality a n (t) n s ≥ a n−ℓ(y) (t) n−ℓ(y) s−1, s ∈ [0, n−ℓ(y)], which implies that (5.14) (Note that we have also used the fact that the law of { L n τ(t) (x) : x ∈ T ≤n−ℓ(y) } is the same as that of { L n−ℓ(y) Fix v ∈ T n−ℓ(y) . Recall the definitions of L ↓ and τ ↓ (·) from the beginning of the proof. By Lemma 2.1, the third term on the right-hand side of (5.13) is bounded from above by We estimate the indicator function in the expectation in (5.16) from above by Let H i , i ∈ {1, 2, 3} be the expectation obtained from the one in (5.16) by replacing the indicator function in it with 1 E i . In particular, the right of (5.16) is bounded from above by b n−ℓ(y) ( To estimate H 1 , we use Proposition 3.1(i) and the density of B n−ℓ(y) . To estimate H 2 , we use Proposition 3.1(i) and (3.28). To estimate H 3 , we use (3.28) and bound the probability in H 3 from above just by 1. Taking n 0 = n 0 (y, ℓ(y)) ∈ N large enough, for all n ≥ n 0 , (5.16) is bounded from above by where ε n , n ≥ 0 is a sequence with ε n → 0 as n → ∞. By (5.13) and (5.17) together with Proposition 3.1(ii) and Lemma 3.2, taking n 0 = n 0 (y, ℓ(y)) ∈ N large enough, we have for all n ≥ n 0 E ρ ( Λ n y,ℓ(y) (t)) ≥ c 5 ye −2 To obtain a lower bound of P ρ (max v∈T n L n τ(t) (v) ≥ √ t + a n (t) + y)/ E ρ [Λ n y,ℓ(y) (t)], we need the following: y 1/20 , there exists n 0 = n 0 (y, ℓ(y)) ∈ N such that for all n ≥ n 0 and t ≥ c 1 n log n, Proof. Fix any y ≥ y 0 and ℓ(y) > e 8 √ log b 3 y 1/20 , where we take y 0 > 0 large enough. Throughout the proof, given n ∈ N, we assume t ≥ c * n log n for some sufficiently large c * > 0. We have time of a Brownian motion on T v∧w ≤ℓ(y)+k and set τ ↓ (s) := inf{r ≥ 0 : L ↓ r (v ∧ w) > s}. By Lemma 2.1, P ρ (F n v,y,ℓ(y) (t) ∩ F n w,y,ℓ(y) (t)) is equal to where for each x ∈ {v, w}, Under the assumption (5.22), we estimate the probabilities in (5.21). Fix x ∈ {v, w}. Let L ↓↓ be a local time of a Brownian motion on T x ≤ℓ(y) . We define the inverse local time by τ ↓↓ (s) := inf{r ≥ 0 : L ↓↓ r (x) > s}. By Lemma 2.1, P x is equal to We use the following estimate in the integrand of (5.25): for each ⌊y⌋ ≤ k ≤ n − ℓ(y) − 1. By this and Proposition 3.1(i), the right-hand side of (5.25) is bounded from above by Recall the events in the indicator function in (5.21) and in (5.22). We estimate the probability of the intersection of these events. Using Lemma 2.3, (2.6), and and the change of measure (3.15)-(3.16) for n ≥ n 0 (n 0 = n 0 (y, ℓ(y)) ∈ N large enough), we have (5.27) By (2.11), the right of (5.27) is bounded from above by We will use the following in the integrand of (5.28): e By this, the right-hand side of (5.28) is bounded from above by where We divide the sum over 1 ≤ k < n − ℓ(y) in (5.20) into sums over the following: (a) 1 ≤ k ≤ ⌊y⌋; (b) ⌊y⌋ < k ≤ ⌊(n − ℓ(y))/2⌋; (c) ⌊(n − ℓ(y))/2⌋ < k ≤ n − ℓ(y) − ⌊y⌋; (d) n − ℓ(y) − ⌊y⌋ < k ≤ n − ℓ(y) − 1. We make remarks on how to estimate the sums over difficult regimes, (b) and (c): In the regime (b), we use the fact that k − 3 2 e 3k log n 2n is bounded from above by a universal constant. In the regime of (c), we use the estimate , for some universal constant c. By (5.21), (5.26), and (5.29), we have In the case k = n − ℓ(y), by the independence of excursions of a Brownian motion around ρ, we have We will obtain upper and lower bounds of E ρ [Λ n y,ℓ(y) (t)]. By (5.12) and (5.18), taking n 0 = n 0 (y, ℓ(y)) large enough, we have for all n ≥ n 0 By the arguments in (5.4) and (5.5), E ρ [Λ n y,ℓ(y) (t)] is bounded from above by b n−ℓ(y) times the second term of (5.5). By this together with (2.11) and y 1/20 , there exists n 0 = n 0 (y, ℓ(y)) ∈ N such that for all n ≥ n 0 and t ≥ c 1 n log n, P ρ max v∈T n L n τ(t) (v) > √ t + a n (t) + y E ρ Λ n y,ℓ(y) (t) ≤ (1 + δ y ) 1 + e −c 2 y 1/20 + δ y , (5.35) Proof. Fix any y ≥ y 0 , ℓ(y) > e 8 √ log b 3 y 1/20 , n ≥ n 0 , and t ≥ c * n log n, where we take y 0 > 0, n 0 = n 0 (y, ℓ(y)) ∈ N, c * > 0 large enough. We first obtain the upper bound. Recall the event G n−ℓ(y) y+y 1/20 −2 (t) from (3.3) and ε n from (5.17). Recall the inequality in (5.14). We have which proves (5.36).
Proof of Proposition 5.1. Let (h v ) v∈T be a BRW on T defined in Section 1. By Lemma A.7, one can show that the sequence ℓ ℓ 2/5 is bounded from above and away from 0. Fix a nondecreasing sequence (ℓ 0 (y + k )) k≥1 (y + k ) 1/20 for each k ≥ 1. By the boundedness of the sequence (5.43), there exists a subsequence (ℓ 0 (y + k j )) j≥1 of (ℓ 0 (y + k )) k≥1 such that the limit exists. We set Note that by the definition of ℓ 0 (y + j ), for any y j with y j ≤ y + j , we have Fix j ≥ 1 and y j with y − j ≤ y j ≤ y + j . Recall definitions J ℓ j and Λ n y j ,ℓ j ,J ℓ j (t) from (5.39) and (5.38). Fix v * ∈ T n−ℓ j . Let L ↓ be a local time of a Brownian motion on T v * ≤ℓ j . We set τ ↓ (s) := inf{r ≥ 0 : L ↓ r (v * ) > s}. Recall P B n−ℓ j and B from (3.15) and (3.16). We define φ (x) by replacing y, ℓ(y), and v in the definition of ψ(x) in (5.6) by y j , ℓ j , and v * , respectively. By similar arguments to those in (5.4) and (5.5), we have for t ≥ n log n, where we have used the fact that for t ≥ n log n, exp − 3 8 n−ℓ j 0 ds X s = 1 + O((log n) −1 ) under the event that X s /2 ≥ δ √ t for all 0 ≤ s ≤ n − ℓ j . We first estimate K 2 . By (3.28) and Proposition 3.1(i), for all ε > 0 and j ≥ 1, there exists n 0 ( j) = n 0 (y − j , y + j , ℓ j ) ∈ N such that for all n ≥ n 0 ( j), we have uniformly in y j and t satisfying (5.2) (we take c * large enough). Next, we estimate K 1 . By the density (2.11), K 1 is equal to where we have set Here, and ∆ n,t y j ,ℓ j (z) is the remainder term so that P v * [A n,t y j ,ℓ j (z)] = P ρ [B n,t y j ,ℓ j (z)]. (Note that by the symmetry of the b-ary tree, the law of the Brownian motion on T v * ≤ℓ j starting at v * is the same as that of the Brownian motion on T ≤ℓ j starting at ρ.) One can show that for each j ≥ 1, where (h v ) v∈T is a BRW. By this together with the definition of γ * in (5.44) and (5.48), for all ε > 0, there exists j 0 ∈ N such that the following holds: for each j ≥ j 0 , there exists n 0 ( j) = n 0 (y − j , y + j , ℓ j ) ∈ N such that for all n ≥ n 0 ( j), (5.49) uniformly in y j and t satisfying (5.2). Thus, by (5.45)-(5.47), (5.49), and the definition (1.8) of β * , for all ε > 0, there exists j 0 ∈ N such that the following holds: for each j ≥ j 0 , there exists n 0 ( j) = n 0 (y − j , y + j , ℓ j ) ∈ N such that for all n ≥ n 0 ( j), uniformly in y j and t satisfying (5.2). By Lemma 5.4,5.5,and (5.50), for all ε > 0, there exists j 0 ∈ N such that the following holds: for each j ≥ j 0 , there exists n 0 ( j) = n 0 (y − j , y + j , ℓ j ) ∈ N such that for all n ≥ n 0 ( j), (5.1) holds by replacing γ * with γ * uniformly in y j and t satisfying (5.2) (we take c * > 0 large enough). Let γ * be the limit of any convergent subsequence of (5.43). By taking a sub-subsequence, if necessary, and repeating the above argument, we have (5.1) if we replace γ * with γ * . Thus, the full sequence (5.43) converges to a finite positive constant and we write γ * to denote the limit. Therefore, we have (5.1).

Proof of Theorem 1.1 and Corollary 1.3
In this section, we prove Theorem 1.1 and Corollary 1.3. We begin with preliminary lemmas. Let (h v ) v∈T be a BRW on T defined in Section 1. For each n ∈ N, we set D (2) Then the following holds: Fix any y > 0. By the equality 1 {max v∈Tn h v ≤m n +y} + 1 {max v∈Tn h v >m n +y} = 1, we bound the probability P(D (2) n ≥ ε) from above by Since h v is a Gaussian random variable with mean 0 and variance n/2 for each v ∈ T n , a simple calculation implies that By this, Lemma A.7(i), and (6.2), we have lim sup n→∞ P D (2) n ≥ ε ≤ c 1 (1 + y)e −2 √ log b y .
Recall the definition of σ (·) from (1.2). For a ∈ R, an interval I ⊂ R, t > 0, and r, n ∈ N with r < n, we set Proof. Fix any finite interval I ⊂ R. Let {I, I} be the boundary of I with I < I. Fix any t > 0, a ∈ R, r ≥ r 0 (I), and n > r, where we take r 0 (I) > |I| large enough. Recall the event G n r (t) from (3.3). P ρ [ B σ r,t,n (I; a) c ] is bounded from above by and the label (v 1 , . . . , v n ). By this and a simple observation, one can see

This implies
v ∈ T n : σ (v r ) ∈ a − b −r , a ≤ c 1 rb n−r . (6.6) By (6.6) and similar arguments to those in (3.23) and (3.24), the first term on the righthand side of (6.5) is bounded from above by c 2 (I)b −r r 3 . By this and Lemma 3.2, we have (6.4).
Proof of Theorem 1.1. Recall definitions of Z ∞ , Ξ (m) n,t , β * , and γ * from (1.4), (1.5), (1.8), and (1.9). For each interval I ⊂ R, we will write ∂ I to denote its boundary. Fix any sequence of positive integers (r n ) n≥1 with lim n→∞ r n = ∞ and lim n→∞ r n /n ∈ [0, 1). Fix any (t n ) n≥1 with lim n→∞ √ t n /n = θ ∈ [0, ∞] and t n ≥ c * n log n, where c * is a sufficiently large positive constant. By, for example, [25, Proposition 11.1.VIII], in order to show the convergence of the point process Ξ (r n ) n,t n to the Cox process (1.7) as n → ∞, it is enough to prove the following: for all finite disjoint intervals and positive values a 1 , . . . , a m , Let q < n be a positive integer. To show (6.7), we first prove convergence of Ξ (n−q) n,t n as n → ∞, q → ∞. Recall the events (6.3). We have Thus, we have We estimate J 1,1 . By Theorem 2.2, on the same probability space (we will write P to denote the probability measure), we can construct a local time (L n τ(t n ) (v)) v∈T ≤n and two BRWs (h v ) v∈T ≤n , (h ′ v ) v∈T ≤n satisfying (2.4) and (2.5). Fix δ ∈ (0, 1/3). We set For each v ∈ T q , let L ↓ be a local time of a Brownian motion on T v ≤n−q and set τ ↓ (p) := inf{s ≥ 0 : L ↓ s (v) > p}. We omit the subscript v in L ↓ and τ ↓ . By Lemma 2.1, we have where for each v ∈ T q , we have set (6.14) By Lemma A.7, we have On the event C q , by (2.5), we have for all v ∈ T q √ t n + a n (t n ) = L n τ(t n ) (v) + a n−q L n We take any sufficiently small ε > 0 and sufficiently large q 0 ∈ N which depends on B 1 , . . . , B m and ε. We assume that q ≥ q 0 and n ≥ n 0 , where we take sufficiently large n 0 = n 0 (q, ε) ∈ N. By Proposition 5.1 and (6.17)-(6.18), under the event C q , we have for all We set D (2) Recall the random measure Z q from (1.3). By (6.16), (6.19), and Taylor's expansion of We can obtain a similar lower bound of ∏ v∈T q K v . By Lemma A.7, Lemma 6.1, and the fact that lim q→∞ W q = 0 almost surely (see [35]), we have Thus, by the above estimates, taking n → ∞, then q → ∞, and finally ε → 0, we have (6.22) Next, by using (6.22), we will prove (6.7). Let z * be a real number with z * < min 1≤i≤m B i . Take q 0 = q 0 (z * ) ∈ N large enough and fix any q ≥ q 0 . Take n ∈ N large enough so that q < n − r n < n − q and q < n/4. We set U n z * ,q (t n ) := ∃v, u ∈ T n with q ≤ |v ∧ u| ≤ n − q s.t.
Under the event U n z * ,q (t n ) c , we have

A.2 Proof of Lemma 2.1
In this section, we use notation in Section 2. Let X = ( X t ,t ≥ 0, P x , x ∈ T ≤n ) be a Brownian motion on T ≤n . Let { h x : x ∈ T ≤n } be a centered Gaussian process associated with X (see Section A.1 for the definition). In the proof of Lemma 2.1, we will use a variation of Theorem 2.2: ( h x + c) 2 : x ∈ T ≤n under P ρ × P is the same as that of 1 2 ( h x + 2t + c 2 ) 2 : x ∈ T ≤n under P.

A.3 Tail of maximum of local time revisited
In the proof of Proposition 4.1, we need a version of Proposition 3.1(ii): Proposition A.3 There exist c 1 > 0 and t 0 > 0 such that for all n ∈ N, t ≥ t 0 , and y ∈ [0, 2 √ n], P ρ max v∈T n L n τ(t) (v) ≥ √ t + a n (t) + y ≥ c 1 e −2 √ log b y . (A.13) Remark A. 4 The assumption of t in Proposition A.3 is weaker than that of Proposition 3.1(ii). This is the main requirement in the proof of Proposition 4.1.