Bootstrapping Nonequilibrium Stochastic Processes

TL;DR

This work develops a general bootstrap framework for nonequilibrium stochastic processes on infinite lattices by exploiting the positivity of probability measures together with invariance, master equations, and initial conditions to obtain rigorous bounds. It formulates linear programming (and semidefinite programming) hierarchies that bound invariant-measure expectations and time-evolved observables, then applies them to the contact process and Domany-Kinzel models in asynchronous and synchronous settings across 1+1 and 2+1 dimensions. The results yield rigorous lower bounds on critical rates and two-sided bounds on half-lives and temporal correlation lengths, with numerical convergence observed as the hierarchy level $L$ increases. The framework is broadly applicable and offers a path toward sharper bounds and insights into nonequilibrium universality classes, potentially connecting to bootstrap ideas and conformal-type approaches in nonequilibrium contexts.

Abstract

We show that bootstrap methods based on the positivity of probability measures provide a systematic framework for studying both synchronous and asynchronous nonequilibrium stochastic processes on infinite lattices. First, we formulate linear programming problems that use positivity and invariance property of invariant measures to derive rigorous bounds on their expectation values. Second, for time evolution in asynchronous processes, we exploit the master equation along with positivity and initial conditions to construct linear and semidefinite programming problems that yield bounds on expectation values at both short and late times. We illustrate both approaches using two canonical examples: the contact process in 1+1 and 2+1 dimensions, and the Domany-Kinzel model in both synchronous and asynchronous forms in 1+1 dimensions. Our bounds on invariant measures yield rigorous lower bounds on critical rates, while those on time evolutions provide two-sided bounds on the half-life of the infection density and the temporal correlation length in the subcritical phase.

Figures (14)

• Figure 1: Left: $LP_{inv}(L)$ upper bounds on $\rho$ for the contact process on $\mathbb Z$ at $L=7$ (black), $L=8$ (orange), $L=9$ (blue), and $L=10$ (green), and also the KMC estimates (yellow) with $1\sigma$ error bars, which are hardly visible. The estimate for the critical rate $\lambda_c\approx1.6491(1)$ from PhysRevE.63.041109 is marked in red. Right: $LP_{inv}(L)$ upper bounds on $\nu$ for the contact process on $\mathbb Z$, together with the KMC estimates. Colors for data points are identical to those in the left figure.
• Figure 2: $LP_{inv}(L)$ lower bounds on $\lambda_c$ for the contact process on $\mathbb Z$ at different values of $L$ (blue dots). For comparison, the estimate $\lambda_c\approx1.6491(1)$ is also shown (dotted red line).
• Figure 3: Left: $LP_{inv}(L)$ upper bounds on $\rho$ for the asynchronous Domany-Kinzel model at $L=7$ (black), $L=8$ (orange), $L=9$ (blue), and $L=10$ (green), and also the KMC estimates (yellow) with $1\sigma$ error bars, which are hardly visible. The estimate for the critical rate $p_{1c}\approx0.908$ is marked in red. Right: $LP_{inv}(L)$ lower bounds on $p_{1c}$ for the asynchronous Domany-Kinzel model at different values of $L$ (blue dots). For comparison, the estimate $p_{1c}\approx0.908$ is also presented (dotted red line).
• Figure 4: $LP_{inv}^{2d}(L)$ upper bounds on $\rho$ for the contact process on $\mathbb Z^2$ at $L=1$ (black line) and $L=2$ (blue dots), and KMC estimates (orange dots) with $1\sigma$ error bars, which are hardly visible. The Monte Carlo estimate for $\lambda_{c,2}\approx0.41220(3)$ from PhysRevE.54.R3090 is marked in red.
• Figure 5: Left: $LP_{inv}^s(L)$ upper bounds on $\rho$ for the synchronous Domany-Kinzel model at $L=6$ (black), $L=7$ (orange), $L=8$ (blue), and $L=9$ (green), and also the Monte Carlo estimates (yellow) obtained by averaging over 200 independent simulations on a periodic lattice of size 201 for 400 time steps. Monte Carlo $1\sigma$ error bars are hardly visible. The estimate for the critical rate $p_{1c}\approx0.799$Zebende1994 is marked in red. Right: $LP_{inv}^s(L)$ lower bounds on $p_{1c}$ at different values of $L$ (blue dots). For comparison, the estimate $p_{1c}\approx0.799$Zebende1994 is also presented (dotted red line).

Theorems & Definitions (12)

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