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Daisuke Shiraishi

Publications and source records attributed to Daisuke Shiraishi.

At least 19 recordsLinked to original sources
arXiv2026

Graph distance and effective resistance of the random walk trace in four and five dimensions

In this paper, we prove that the fluctuations of the graph distance and the effective resistance on the trace of a random walk in four and five dimensions converge in distribution to a stable law. In previous work, the first and second authors proved that the corresponding fluctuations converge to a Gaussian distribution in dimensions six and higher. Taken together, these results reveal a phase transition between dimensions five and six. Our proof develops a novel coupling with long range percolation, and we expect this technique to find applications in a broad class of related models.

math.PR↗
arXiv2026

Collision properties of the four-dimensional random walk trace

We consider collisions of multiple random walks on the trace of a simple random walk on the four-dimensional integer lattice. For two independent walks (in continuous time), we apply a result of Noda to derive a scaling limit for the collision time process. For three independent walks (in discrete time), we demonstrate infinitely many triple collisions occur.

math.PR↗
arXiv2026

Tail exponents of the three-dimensional uniform spanning tree and Abelian sandpile

We study the local geometry of the three-dimensional uniform spanning tree and its connection with the Abelian sandpile model. We obtain sharp tail exponents, up to subpolynomial errors, for the past of the origin in the three-dimensional UST and for the $0$-tree of the $0$-wired uniform spanning forest. As a principal application, we prove the corresponding three-dimensional Abelian sandpile avalanche exponents: the avalanche-cluster radius has tail exponent $1$, while both the avalanche-cluster size and the total number of topplings have tail exponent $1/3$. These results identify the leading power-law behaviour of three-dimensional sandpile avalanches and improve previously known bounds.

math.PR↗
arXiv2026

Largest-loop-first loop-erased random walk on $\mathbb{Z}^{4}$

Let $S = (S(n))$ be a simple random walk on $\mathbb{Z}^{d}$ started at the origin. We study a loop-erasing procedure of $S[0,n]$ that differs from Lawler's chronological loop-erasure. Specifically, we remove loops from $S[0,n]$ in decreasing order of their lengths. The resulting random simple path is called the largest-loop-first (LLF) LERW. For $d=4$, we prove that the expected length of LLF LERW is of the order $n (\log n)^{-1/2 + o(1)}$. In particular, this suggests that chronological LERW and LLF LERW belong to different universality classes. Furthermore, we also prove the convergence of LLF LERW to Brownian motion in four dimensions.

math.PR↗
arXiv2025

Sharp one-point estimates and Minkowski content for the scaling limit of three-dimensional loop-erased random walk

In this work, we consider the scaling limit of loop-erased random walk (LERW) in three dimensions and prove that the limiting occupation measure is equivalent to its $β$-dimensional Minkowski content, where $β\in (1, 5/3]$ is its Hausdorff dimension. In doing this we also establish the existence of the two-point function and provide some sharp estimates on one-point function and ball-hitting probabilities for 3D LERW in any scale, which is a considerable improvement of previous results.

math.PR↗
arXiv2024

Convergence of three-dimensional loop-erased random walk in the natural parametrization

In this work we consider loop-erased random walk (LERW) and its scaling limit in three dimensions, and prove that 3D LERW parametrized by renormalized length converges to its scaling limit parametrized by some suitable measure with respect to the uniform convergence topology in the lattice size scaling limit. Our result greatly improves the work (Acta Math. 199(1):29-152) of Gady Kozma which establishes the weak convergence of the rescaled trace of 3D LERW towards a random compact set with respect to the Hausdorff distance.

math.PR↗
arXiv2023

Annealed transition density of simple random walk on a high-dimensional loop-erased random walk

We derive sub-Gaussian bounds for the annealed transition density of the simple random walk on a high-dimensional loop-erased random walk. The walk dimension that appears in these is the exponent governing the space-time scaling of the process with respect to the extrinsic Euclidean distance, which contrasts with the exponent given by the intrinsic graph distance that appears in the corresponding quenched bounds. We prove our main result using novel pointwise Gaussian estimates on the distribution of the high-dimensional loop-erased random walk.

math.PR↗
arXiv2023

Scaling limit of the occupation measure of random walk cut points

We consider the occupation measure of the cut points of a simple random walk on a $d$-dimensional cubic lattice for $d = 2, 3$, and we show that the scaling limit of the occupation measure in weak topology is the natural fractal measure on the Brownian cut points defined via its Minkowski content.

math.PR↗
arXiv2023

On the size of earthworm's trail

We investigate the number of holes created by an ``earthworm'' moving on the two-dimensional integer lattice. The earthworm is modeled by a simple random walk. At the initial time, all vertices are filled with grains of soil except for the position of the earthworm. At each step, the earthworm pushes the soil in the direction of its motion. It leaves a hole (an empty vertex with no grain of soil) behind it. If there are holes in front of the earthworm (in the direction of its step), the closest hole is filled with a grain of soil. Thus the number of holes increases by 1 or remains unchanged at every step. We show that the number of holes is at least $\mathcal{O}(n^{3/4})$ after $n$ steps.

math.PR↗
arXiv2021

Scaling limit for random walk on the range of random walk in four dimensions

We establish scaling limits for the random walk whose state space is the range of a simple random walk on the four-dimensional integer lattice. These concern the asymptotic behaviour of the graph distance from the origin and the spatial location of the random walk in question. The limiting processes are the analogues of those for higher-dimensional versions of the model, but additional logarithmic terms in the scaling factors are needed to see these. The proof applies recently developed machinery relating the scaling of resistance metric spaces and stochastic processes, with key inputs being natural scaling statements for the random walk's invariant measure, the associated effective resistance metric, the graph distance, and the cut times for the underlying simple random walk.

math.PR↗
arXiv2021

Scaling limits of the three-dimensional uniform spanning tree and associated random walk

We show that the law of the three-dimensional uniform spanning tree (UST) is tight under rescaling in a space whose elements are measured, rooted real trees, continuously embedded into Euclidean space. We also establish that the relevant laws actually converge along a particular scaling sequence. The techniques that we use to establish these results are further applied to obtain various properties of the intrinsic metric and measure of any limiting space, including showing that the Hausdorff dimension of such is given by $3/β$, where $β\approx 1.624\dots$ is the growth exponent of three-dimensional loop-erased random walk. Additionally, we study the random walk on the three-dimensional uniform spanning tree, deriving its walk dimension (with respect to both the intrinsic and Euclidean metric) and its spectral dimension, demonstrating the tightness of its annealed law under rescaling, and deducing heat kernel estimates for any diffusion that arises as a scaling limit.

math.PR↗
arXiv2020

The number of spanning clusters of the uniform spanning tree in three dimensions

Let ${\mathcal U}_δ$ be the uniform spanning tree on $δ\mathbb{Z}^{3}$. A spanning cluster of ${\mathcal U}_δ$ is a connected component of the restriction of ${\mathcal U}_δ$ to the unit cube $[0,1]^{3}$ that connects the left face $\{ 0 \} \times [0,1]^{2}$ to the right face $\{ 1 \} \times [0,1]^{2}$. In this note, we will prove that the number of the spanning clusters is tight as $δ\to 0$, which resolves an open question raised by Benjamini (1999).

math.PR↗
arXiv2018

One-point function estimates for loop-erased random walk in three dimensions

In this work, we consider loop-erased random walk (LERW) in three dimensions and give an asymptotic estimate on the one-point function for LERW and the non-intersection probability of LERW and simple random walk in three dimensions for dyadic scales. These estimates will be crucial to the characterization of the convergence of LERW to its scaling limit in natural parametrization. As a step in the proof, we also obtain a coupling of two pairs of LERW and SRW with different starting points conditioned to avoid each other.

math.PR↗
arXiv2016

Hausdorff dimension of the scaling limit of loop-erased random walk in three dimensions

Let $M_{n}$ be the length (number of steps) of the loop-erasure of a simple random walk up to the first exit from a ball of radius $n$ centered at its starting point. It is shown in [18] that there exists $β\in (1, \frac{5}{3}]$ such that $E (M_{n} )$ is of order $n^β$ in 3 dimensions. In the present article, we show that the Hausdorff dimension of the scaling limit of the loop-erased random walk in 3 dimensions is equal to $β$ almost surely.

math.PR↗