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

TL;DR

The paper analyzes diffusion-limited pair annihilation and deposition (DLPAD) on a finite 1D lattice under conditioning on atypical activity. By mapping the tilted dynamics to a quantum XY chain, it obtains all cumulants of the activity to leading order in the system size $L$ at the critical point, where the conditioned process exhibits an Ising-type dynamical phase transition with central charge $c=1/2$ and ballistic scaling. The authors establish a universal cumulant generating function $h(u)$ that governs the finite-size scaling of cumulants near criticality, showing that $\lim_{L\to\infty} L^2 K_0(L,u) = \frac{\pi c}{6} + h(\sqrt{\xi^2-1}\,u)$ with $\xi = 1/\sqrt{1-\nu^2}$ and $c=1/2$, thereby linking universal large-scale behaviour to conformal invariance. They also provide explicit asymptotics for mean, variance, skewness, and higher cumulants, including detailed expressions for the universal scaling coefficients in terms of the microscopic parameter $\nu$ and zeta-function data, highlighting a distinct universality class from other models like SSEP. This work strengthens the connection between dynamical large deviations, conformal field theory, and exactly solvable spin chains, offering a rigorous framework for exploring other universality classes and boundary conditions in conditioned stochastic dynamics.

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.

Figures (1)

• Figure 1: Blue curve: The scaling function $h(\sqrt{\xi^2-1} \, u)$ for $\nu=0.9$. The values at the vertical lines are numerical evaluations of the exact generating function for different L. They all collapse onto the same point along the blue curve.

Theorems & Definitions (16)

• Definition 2.1
• Remark 2.2
• Lemma 3.1
• Lemma 3.2
• Lemma 3.3
• Proposition 3.4
• Proposition 3.5
• Remark 3.6
• Proposition 3.7
• Proposition 3.8