Universal and non-universal large deviations in critical systems

TL;DR

This paper investigates the statistical behavior of rare events in critical, scale-invariant systems, revealing both universal and nonuniversal features of their probability distribution tails through various analytical and numerical methods.

Contribution

It introduces a comprehensive analysis of tail behaviors in critical systems, extending Cramér's series to correlated variables and highlighting the crossover from universal to nonuniversal regimes.

Findings

Identification of universal tail behaviors in critical $O(n)$ systems

Extension of Cramér's series to strongly correlated variables

Challenging existing assumptions about power-law corrections in tail decay

Abstract

Rare events play a crucial role in understanding complex systems. Characterizing and analyzing them in scale-invariant situations is challenging due to strong correlations. In this work, we focus on characterizing the tails of probability distribution functions (PDFs) for these systems. Using a variety of methods, perturbation theory, functional renormalization group, hierarchical models, large $n$ limit, and Monte Carlo simulations, we investigate universal rare events of critical $O(n)$ systems. Additionally, we explore the crossover from universal to nonuniversal behavior in PDF tails, extending Cram\'er's series to strongly correlated variables. Our findings highlight the universal and nonuniversal aspects of rare event statistics and challenge existing assumptions about power-law corrections to the leading stretched exponential decay in these tails.