Condensation transition in joint large deviations of linear statistics

Abstract

Abstract. Real space condensation is known to occur in stochastic models of mass transport in the regime in which the globally conserved mass density is greater than a critical value. It has been shown within models with factorised stationary states that the condensation can be understood in terms of sums of independent and identically distributed random variables: these exhibit condensation when they are conditioned to a large deviation of their sum. It is well understood that the condensation, whereby one of the random variables contributes a finite fraction to the sum, occurs only if the underlying probability distribution (modulo exponential) is heavy-tailed, i.e. decaying slower than exponential. Here we study a similar phenomenon in which condensation is exhibited for non-heavy-tailed distributions, provided random variables are additionally conditioned on a large deviation of certain linear statistics. We provide a detailed theoretical analysis explaining the phenomenon, which is supported by Monte Carlo simulations (for the case where the additional constraint is the sample variance) and demonstrated in several physical systems. Our results suggest that the condensation is a generic phenomenon that pertains to both typical and rare events.

Conditioned random walks and interaction-driven condensation

“…This condensation mechanism is very similar to the one we previously studied in Ref. [23,24]. There the partition function of random variables m i , i = 1, .…”

Anomalous Scaling of Dynamical Large Deviations

“…Here, we find "temporal condensates" in the form of trajectories for the fluctuations of A τ that are localized in time compared to τ and whose height scales with τ . A related condensation was reported recently in the context of sums of random variables, which can be dominated in some cases by a single, extensive or "giant" value [69][70][71][72][73][74].…”

Condensation of degrees emerging through a first-order phase transition in classical random graphs

“…The term condensation of fluctuations has been coined to describe such condensed states that are triggered by large deviations of an extensive random variable, but whose typical behaviour does not necessarily show any sign of condensation. Examples of random systems, where condensation emerges as a rare event, include the Gaussian model [5], the Urn model [6], models of mass transport [7,8], to name just a few. Condensation of fluctuations usually brings about a rich phenomenology including phase transitions, giant responses to small perturbations, and singularities in the full probability distribution [9].…”