Density profiles, dynamics, and condensation in the ZRP conditioned on an atypical current

TL;DR

This paper analyzes the zero-range process conditioned on atypical currents, revealing non-trivial density profiles and condensate formation, and connects microscopic results with macroscopic fluctuation theory predictions.

Contribution

It explicitly computes the effective process for the conditioned ZRP, showing space-dependent rates and validating MFT predictions in the weakly asymmetric regime.

Findings

Effective process has space-dependent hopping rates.

Conditioned dynamics exhibit non-trivial density profiles.

Condensates can form in a supercritical bulk region.

Abstract

We study the asymmetric zero-range process (ZRP) with L sites and open boundaries, conditioned to carry an atypical current. Using a generalized Doob h-transform we compute explicitly the transition rates of an effective process for which the conditioned dynamics are typical. This effective process is a zero-range process with renormalized hopping rates, which are space dependent even when the original rates are constant. This leads to non-trivial density profiles in the steady state of the conditioned dynamics, and, under generic conditions on the jump rates of the unconditioned ZRP, to an intriguing supercritical bulk region where condensates can grow. These results provide a microscopic perspective on macroscopic fluctuation theory (MFT) for the weakly asymmetric case: It turns out that the predictions of MFT remain valid in the non-rigorous limit of finite asymmetry. In addition, the microscopic results yield the correct scaling factor for the asymmetry that MFT cannot predict.