Related papers: Off-Criticality and the Massive Brownian Loop Soup

Several classical results on boundary crossing probabilities of Brownian motion and random walks are extended to asymptotically Gaussian random fields, which include sums of i.i.d. random variables with multidimensional indices,…

We consider a class of Gaussian Free Fields denoted by $(g_x)_{x \in {\cal V}_N}$, where $ {\cal V}_N = \{0,1\}^N$ and $N\in \mathbb{Z}_+$. These fields are related to a general class of $N$-dimensional random walks on the hypercube, which…

We study a system of N non-intersecting Brownian motions on a line segment [0,L] with periodic, absorbing and reflecting boundary conditions. We show that the normalized reunion probabilities of these Brownian motions in the three models…

The extension of strongly anisotropic or dynamical scaling to local scale invariance is investigated. For the special case of an anisotropy or dynamical exponent $\theta=z=2$, the group of local scale transformation considered is the…

We prove a simple identity relating the length spectrum of a Riemann surface to that of the same surface with an arbitrary number of additional cusps. Our proof uses the Brownian loop measure introduced by Lawler and Werner. In particular,…

We study the scaling limit of planar loop erased random walk (LERW) on the percolation cluster, with occupation probability $p\geq p_c$. We numerically demonstrate that the scaling limit of planar LERW$_p$ curves, for all $p>p_c$, can be…

We consider the open symmetric exclusion (SEP) and inclusion (SIP) processes on a bounded Lipschitz domain $\Omega$, with both fast and slow boundary. For the random walks on $\Omega$ dual to SEP/SIP we establish: a functional-CLT-type…

This work studies the $O(N)$ Linear Sigma Model on $\mathbb{R}^{2}$ under a scaling dictated by the formal $1/N$ expansion. We show that in the large $N$ limit, correlations decay exponentially fast, where the acquired mass decays…

We introduce and compute the generalized disconnection exponents $\eta_\kappa(\beta)$ which depend on $\kappa\in(0,4]$ and another real parameter $\beta$, extending the Brownian disconnection exponents (corresponding to $\kappa=8/3$)…

Particle approximations for certain nonlinear and nonlocal reaction-diffusion equations are studied using a system of Brownian motions with killing. The system is described by a collection of i.i.d. Brownian particles where each particle is…

Logarithmic finite-size scaling of the O($n$) universality class at the upper critical dimensionality ($d_c=4$) has a fundamental role in statistical and condensed-matter physics and important applications in various experimental systems.…

We study the uniform random graph $\mathsf{C}_n$ with $n$ vertices drawn from a subcritical class of connected graphs. Our main result is that the rescaled graph $\mathsf{C}_n / \sqrt{n}$ converges to the Brownian Continuum Random Tree…

Given a bounded Riemann surface $M$ of finite topological type, we show the existence of a universal and conformally invariant scaling limit for the Temperleyan cycle-rooted spanning forest on any sequence of graphs which approximate $M$ in…

The multiplicities of the decomposition of the product of an arbitrary number $n$ of spin $s$ states into irreducible $SU(2)$ representations are computed. Two complementary methods are presented, one based on random walks in representation…

We establish an invariance principle connecting boundary random walks on $\mathbb N$ with Feller's Brownian motions on $[0,\infty)$. A Feller's Brownian motion is a Feller process on $[0,\infty)$ whose excursions away from the boundary $0$…

Large-N multi-matrix loop equations are formulated as quadratic difference equations in concatenation of gluon correlations. Though non-linear, they involve highest rank correlations linearly. They are underdetermined in many cases.…

We consider discrete Gaussian free fields with ergodic random conductances on a class of random subgraphs of $\mathbb{Z}^{d}$, $d \geq 2$, including i.i.d.\ supercritical percolation clusters, where the conductances are possibly unbounded…

In these Notes, a comprehensive description of the universal fractal geometry of conformally-invariant scaling curves or interfaces, in the plane or half-plane, is given. The present approach focuses on deriving critical exponents…

We study the quantum critical phase of a SU(2) symmetric spin-2 chain obtained from spin-2 bosons in a one-dimensional lattice. We obtain the scaling of the entanglement entropy and finite-size energies by exact diagonalization and…

We show that almost any one-dimensional projection of a suitably scaled random walk on a hypercube, inscribed in a hypersphere, converges weakly to an Ornstein-Uhlenbeck process as the dimension of the sphere tends to infinity. We also…