Related papers: Hausdorff dimension of the scaling limit of loop-e…

We show that, for any $0<\gamma<1/2$, any $(\alpha,\beta)\in\mathbb{R}^2$ except on a set with Hausdorff dimension about $\sqrt{\gamma}$, any small $0<\varepsilon<1$ and any large $N\in\mathbb{N}$, the number of integers $n\in[1,N]$ such…

Let $\{X(t) : t \in \mathbb{R}^d \}$ be a multivariate operator-self-similar random field with values in $\mathbb{R}^m$. Such fields were introduced in [24] and satisfy the scaling property $\{X(c^E t) : t \in \mathbb{R}^d \} \stackrel{\rm…

For any $\beta>1$, let $T_\beta$ be the classical $\beta$-transformations. Fix $x_0\in[0,1]$ and a nonnegative real number $\hat{v}$, we compute the Hausdorff dimension of the set of real numbers $x\in[0,1]$ with the property that, for…

For the perimeter length and the area of the convex hull of the first $n$ steps of a planar random walk, we study $n \to \infty$ mean and variance asymptotics and establish non-Gaussian distributional limits. Our results apply to random…

We study the scaling limit of essentially simple triangulations on the torus. We consider, for every $n\geq 1$, a uniformly random triangulation $G_n$ over the set of (appropriately rooted) essentially simple triangulations on the torus…

The loop-erased random walk (LERW) in $\mathbb{Z}^4$ is the process obtained by erasing loops chronologically for simple random walk. We prove that the escape probability of the LERW renormalized by $(\log n)^{\frac{1}{3}}$ converges almost…

Let \zeta be the intersection exponent of random walks in Z^3 and \alpha be a positive real number. We construct a stochastic process from a simple random walk by erasing loops of length at most N^\alpha. We will prove that for \alpha <…

We consider loop-erased random walk (LERW) running between two boundary points of a square grid approximation of a planar simply connected domain. The LERW Green's function is the probability that the LERW passes through a given edge in the…

In this article we study a \emph{non-directed polymer model} on $\mathbb Z$, that is a one-dimensional simple random walk placed in a random environment. More precisely, the law of the random walk is modified by the exponential of the sum…

Let $\theta$ be an irrational number and $\varphi: {\mathbb N} \to {\mathbb R}^{+}$ be a monotone decreasing function tending to zero. Let $$E_\varphi(\theta) =\Big\{y \in \mathbb R: \|n\theta- y\|<\varphi(n), \ {\text{for infinitely…

It is shown that the Hausdorff dimension of the fast escaping set of a quasiregular self-map of ${\mathbb R}^3$ can take any value in the interval $[1,3]$. The Hausdorff dimension of the Julia set of such a map is estimated under some…

The Hausdorff dimension of the set of points that are covered infinitely many times by a sequence of randomly distributed balls in the unit cube can be expressed in terms of the sizes of the balls. This note presents a new proof of the…

Let $X = G/\Gamma$, where $G$ is a Lie group and $\Gamma$ is a uniform lattice in $G$, and let $O$ be an open subset of $X$. We give an upper estimate for the Hausdorff dimension of the set of points whose trajectories escape $O$ on average…

We consider the escaping parameters in the family $\beta\wp_\Lambda$, i.e. these parameters for which the orbits of critical values of $\beta\wp_\Lambda$ approach infinity, where $\wp_\Lambda$ is the Weierstrass function. Unlike to the…

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…

We study the random walk representation of the two-point function in statistical mechanics models near the critical point. Using standard scaling arguments we show that the critical exponent $\nu$ describing the vanishing of the physical…

Random walks of n steps taken into independent uniformly random directions in a d-dimensional Euclidean space (d larger than 1), are named Dirichlet when their step lengths are distributed according to a Dirichlet law. The latter continuous…

Let $\beta>1$, $I$ be the unite interval $[0,1)$ and $\phi$ be an integer function defined on $\mathbb{N}\setminus\{0\}$ satisfying $1\leq\phi(n)\leq n$. Denote by $A_\phi(x,\beta)$ the Erd\"{o}s-R\'{e}nyi average of $x\in I$ associated…

We consider a specific random graph which serves as a disordered medium for a particle performing biased random walk. Take a two-sided infinite horizontal ladder and pick a random spanning tree with a certain edge weight $c$ for the…

We consider the Random Walk Pinning Model studied in [3,2]: this is a random walk X on Z^d, whose law is modified by the exponential of \beta times L_N(X,Y), the collision local time up to time N with the (quenched) trajectory Y of another…