A Field-Theoretic Approach to the Wiener Sausage

Abstract

The Wiener Sausage, the volume traced out by a sphere attached to a Brownian particle, is a classical problem in statistics and mathematical physics. Initially motivated by a range of field-theoretic, technical questions, we present a single loop renormalised perturbation theory of a stochastic process closely related to the Wiener Sausage, which, however, proves to be exact for the exponents and some amplitudes. The field-theoretic approach is particularly elegant and very enjoyable to see at work on such a classic problem. While we recover a number of known, classical results, the field-theoretic techniques deployed provide a particularly versatile framework, which allows easy calculation with different boundary conditions even of higher momenta and more complicated correlation functions. At the same time, we provide a highly instructive, non-trivial example for some of the technical particularities of the field-theoretic description of stochastic processes, such as excluded volume, lack of translational invariance and immobile particles. The aim of the present work is not to improve upon the well-established results for the Wiener Sausage, but to provide a fieldtheoretic approach to it, in order to gain a better understanding of the field-theoretic obstacles to overcome.

Volume explored by a branching random walk on general graphs

“…While the scaling behaviour does not depend on the initial position x 0 of a walker, provided it is located in the bulk and remains there as the thermodynamic limit is taken, the field theory has to be adjusted to account for more complicated boundary conditions [5] or the walker starting close to any such boundary. It may also be interesting to consider the case of initialising each site with an independent Poisson distributions of walkers [35].…”

“…where the density of tracers particles at position x and time t > 0 is measured by ψ(x, t) and integrated over all space. Similarly [5], higher moments are determined by integrals of the form a p (t, L) (9) or equivalently, by evaluating the Fourier transform at spatial momentum k = 0. These are functions of the couplings introduced above, but to leading order not of the walker's initial position x 0 , provided it is located in the bulk.…”

“…A virus may multiply at a host cell and then infect any of the neighbouring ones at random [3]. The total number of infected cells therefore corresponds to the number of distinct sites visited by a branching random walk (BRW) [4], also referred to as the Branching Wiener Sausage [5,6]. In this process active random walkers spontaneously produce descendants that carry on hopping from site to site.…”

Field theory of survival probabilities, extreme values, first passage times, and mean span of non-Markovian stochastic processes

“…As is explained in great detail in [35][36][37], and briefly discussed in App. A, the visit probability Q(x, t) can be expressed as a field-theoretic expectation value under a Doi-Peliti field theory by introducing two additional auxiliary ("trace"-) fields ψ(x, t) and ψ(x, t) with a joint distribution…”

“…describing a field with vanishing mass ε > 0 which regularises the infrared, ensures causality [35,38], and is to be taken to ε → 0 + at the end of any calculation. The deposition action is derived in [36] and reads…”