Large deviations, condensation and giant response in a statistical system

Abstract

We study the probability distribution P of the sum of a large number of non-identically distributed random variables nm. Condensation of fluctuations, the phenomenon whereby one of such variables provides a macroscopic contribution to the global probability, is discussed and interpreted in analogy to phase-transitions in Statistical Mechanics. A general expression for P is derived, and its sensitivity to the details of the distribution of a single nm is worked out. These general results are verified by the analytical and numerical solution of some specific examples.

Probability Distributions with Singularities

“…This setting is appropriate to describe a wealth of situations in many areas of science, from network dynamics to financial data. The probability distribution of the total number of particles was studied for large M in different contexts [ 14 , 17 , 21 , 22 , 23 , 43 ]. The (negative) rate function is shown in Figure 3 .…”

“…It has been found that, in many cases, exhibits a singular behavior, in that it is non-differentiable around some value (or values) of the fluctuating variable [ 3 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 ]. Such singularities have an origin akin to those observed in the thermodynamic potentials of systems at criticality.…”

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].…”

Dynamics of fluctuations in the Gaussian model with conserved dynamics

“…In this work we consider the fluctuations dynamics in the Gaussian model, a standard statistical mechanical model which may be regarded as the simplest Ginzburg-Landau theory for the description of the disordered phase of Ising-like systems [21]. In this approach, the probability distribution of the order parameter variance S, the observable we focus on, displays a singular point in S = S c both in and out of equilibrium [4,22,[22][23][24][25][26][27][28][29][30]. This fact is associated to the so-called condensation of fluctuations [22], a phenomenon whereby certain fluctuations are realized by a condensation mechanism in which a single degree of freedom becomes macroscopically populated, similarly to what happens in usual condensation phenomena, e.g., in the Bose gas.…”