arXiv subjects
Publications and source records attributed to Satomi Watanabe.
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.
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.
Refining previous results, we establish a sharp asymptotic estimate on the expected graph distance between the origin and the terminal point of the trace of the first $n$ steps of the walk. A similar conclusion is drawn for the resistance metric.
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.
Let $\mathcal{U}$ be the uniform spanning tree on $\mathbb{Z}^3$, whose probability law is denoted by $\mathbf{P}$. For $\mathbf{P}$-a.s. realization of $\mathcal{U}$, the recurrence of the the simple random walk on $\mathcal{U}$ is proved in [5] and it is also demonstrated in [8] that two independent simple random walks on $\mathcal{U}$ collide infinitely often. In this article, we will give a quantitative estimate on the number of collisions of two independent simple random walks on $\mathcal{U}$, which provides another proof of the infinite collision property of $\mathcal{U}$.
Let $\mathcal{U}$ be the uniform spanning tree on $\mathbb{Z}^{3}$. We show the occurrence of log-logarithmic fluctuations around the leading order for the volume of intrinsic balls in $\mathcal{U}$. As an application, we obtain similar fluctuations for the quenched heat kernel of the simple random walk on $\mathcal{U}$.