This paper illustrates that the bootstrap of a partial sum given by i.i.d. copies of a random variable $X_1$ has to be dealt with care in general. It turns out that in various cases a whole spectrum of different limit…

Consider critical bond percolation on a large $2 n \times 2 n$ box on the square lattice. It is well-known that the size (i.e. number of vertices) of the largest open cluster is, with high probability, of order $n^2 \pi(n)$, where $\pi(n)$ denotes…

The understanding of site percolation on the triangular lattice progressed greatly in the last decade. Smirnov proved conformal invariance of critical percolation, thus paving the way for the construction of its scaling limit. Recently, the scaling limit of near critical…

Constrained diffusions in convex polyhedral cones with a general oblique reflection field, and with a diffusion coefficient scaled by a small parameter $\varepsilon> 0$, are considered. Using an interior Dirichlet heat kernel lower bound estimate for second order elliptic operators in bounded…

Let $X_{1},X_{2},...$ denote independent, identically distributed random variables with common distribution $F$, and $S$ the corresponding random walk with $\rho :=\lim_{n\rightarrow \infty }P(S_{n}>0)$ and $\tau :=\inf \{n\geq 1:S_{n}\leq 0\}$. We assume that $X$ is in the domain of attraction of…

Dyson's Brownian motion model with the parameter $\beta=2$, which we simply call the Dyson model in the present paper, is realized as an $h$-transform of the absorbing Brownian motion in a Weyl chamber of type A. Depending on initial configuration with a finite number of…

The fractional Poisson field (fPf) is constructed by considering the number of balls falling down on each point of $\mathbb R^D$, when the centers and the radii of the balls are thrown at random following a Poisson point process in…

The space of metric measure spaces (complete separable metric spaces with a probability measure) is becoming more and more important as state space for stochastic processes. Of particular interest is the subspace of (continuum) metric measure trees.  Greven, Pfaffelhuber and Winter introduced the Gromov-Prohorov…

Recently, the Kahn-Kalai-Linial (KKL) Theorem on influences of functions on $\{0,1\}^n$ was extended to the setting of functions on Schreier graphs.  Specifically, it was shown that for an undirected Schreier graph $\text{Sch}(G,X,U)$ with log Sobolev constant $\rho$ and generating set…