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We give a simplified proof for the equivalence of loop-erased random walks to a lattice model containing two complex fermions, and one complex boson. This equivalence works on an arbitrary directed graph. Specifying to the $d$-dimensional hypercubic lattice, at large scales this theory reduces to a scalar $phi^4$-type theory with two complex fermions, and one complex boson. While the path integral for the fermions is the Berezin integral, for the bosonic field we can either use a complex field $phi(x)in mathbb C$ (standard formulation) or a nilpotent one satisfying $phi(x)^2 =0$. We discuss basic properties of the latter formulation, which has distinct advantages in the lattice model.
We discuss properties of dipolar SLE(k) under conditioning. We show that k=2, which describes continuum limits of loop erased random walks, is characterized as being the only value of k such that dipolar SLE conditioned to stop on an interval coincid
We investigate the behavior of nonequilibrium phase transitions under the influence of disorder that locally breaks the symmetry between two symmetrical macroscopic absorbing states. In equilibrium systems such random-field disorder destroys the phas
We introduce a diffusion model for energetically inhomogeneous systems. A random walker moves on a spin-S Ising configuration, which generates the energy landscape on the lattice through the nearest-neighbors interaction. The underlying energetic env
We study a model of interacting run-and-tumble random walkers operating under mutual hardcore exclusion on a one-dimensional lattice with periodic boundary conditions. We incorporate a finite, Poisson-distributed, tumble duration so that a particle r
Finite-$N$ effects unavoidably drive the long-term evolution of long-range interacting $N$-body systems. The Balescu-Lenard kinetic equation generically describes this process sourced by ${1/N}$ effects but this kinetic operator exactly vanishes by s