The main theme of these lecture notes is to analyze heat conduction on disordered media
such as fractals and percolation clusters by means of both probabilistic and analytic …

Assume that there is some analytic structure, a differential equation or a stochastic process
for example, on a metric space. To describe asymptotic behaviors of analytic objects, the …

The first main result of this paper is that the law of the (rescaled) two-dimensional uniform
spanning tree is tight in a space whose elements are measured, rooted real trees …

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 …

We present on-diagonal heat kernel estimates and quantitative homogenization statements
for the one-dimensional Bouchaud trap model. The heat kernel estimates are obtained using …

We construct critical percolation clusters on the diamond hierarchical lattice and show that
the scaling limit is a graph directed random recursive fractal. A Dirichlet form can be …

In this article, we study a simple random walk on a decorated Galton-Watson tree, obtained
from a Galton-Watson tree by replacing each vertex of degree $ n $ with an independent …

It is well known that stochastic processes on fractal spaces or in certain random media
exhibit anomalous heat kernel behaviour. One manifestation of such irregular behaviour is …

Consider a family of random ordered graph trees (Tn) n≥ 1, where Tn has n vertices. It has
previously been established that if the associated search-depth processes converge to the …

A scaling limit for the simple random walk on the largest connected component of the Erdos–
Rényi random graph G (n, p) in the critical window, p= n− 1+ λn− 4/3, is deduced. The …