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Daisuke Shiraishi

Publications and source records attributed to Daisuke Shiraishi.

22 records · Page 2Linked to original sources
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↗
arXiv2015

On Brownian motion, simple paths, and loops

We provide a decomposition of the trace of the Brownian motion into a simple path and an independent Brownian soup of loops that intersect the simple path. More precisely, we prove that any subsequential scaling limit of the loop erased random walk is a simple path (a new result in three dimensions), which can be taken as the simple path of the decomposition. In three dimensions, we also prove that the Hausdorff dimension of any such subsequential scaling limit lies in $(1,\frac53]$. We conjecture that our decomposition characterizes uniquely the law of the simple path. If so, our results would give a new strategy to the existence of the scaling limit of the loop erased random walk and its rotational invariance.

math.PR↗