Universal cumulants and conformal invariance in annihilating random walks with pair deposition

TL;DR

This paper analyzes annihilating random walks with pair deposition on a finite one-dimensional torus, deriving universal cumulants of activity at criticality, revealing connections to conformal invariance and the Ising universality class.

Contribution

It provides closed-form expressions for cumulants of activity in a conditioned annihilating random walk at criticality, linking the process to conformal field theory and universal scaling functions.

Findings

Cumulants are obtained in closed form at criticality.

The generating function involves the central charge c=1/2.

Universal scaling functions are identified near the critical point.

Abstract

We consider annihilating random walks on the finite one-dimensional integer torus with deposition of pairs of particles, conditioned on an atypical jump activity. All cumulants of the activity, defined as the number of particle jumps up to some time t, are obtained in closed form to leading order in system size L at the critical point, where in the thermodynamic limit the conditioned process undergoes a phase transition in the universality class of the one-dimensional quantum Ising model in a transverse field. The generating function of the cumulants at a distance of order 1/L away from the critical point is proved to be given by two universal quantities, viz., by the central charge c = 1/2 of the Virasoro algebra that characterizes the Ising universality class and by an explicit universal scaling function.