Reply to Comment on "Towards a large deviation theory for strongly correlated systems"

TL;DR

This paper defends the idea that strongly correlated systems exhibit large deviation behaviors characterized by power-laws and q-exponentials, challenging previous claims and emphasizing the role of Q-Gaussian attractors.

Contribution

It clarifies the connection between strong correlations, q-exponentials, and Q-Gaussian attractors, countering previous analyses that overlooked these aspects.

Findings

Power-law decay replaces exponential law for large N.

Subdominant term aligns with q-exponential behavior.

Q-Gaussian attractors differ from Lévy distributions except for Cauchy-Lorentz.

Abstract

The paper that is commented by Touchette contains a computational study which opens the door to a desirable generalization of the standard large deviation theory (applicable to a set of $N$ nearly independent random variables) to systems belonging to a special, though ubiquitous, class of strong correlations. It focuses on three inter-related aspects, namely (i) we exhibit strong numerical indications which suggest that the standard exponential probability law is asymptotically replaced by a power-law as its dominant term for large $N$; (ii) the subdominant term appears to be consistent with the $q$-exponential behavior typical of systems following $q$-statistics, thus reinforcing the thermodynamically extensive entropic nature of the exponent of the $q$-exponential, basically $N$ times the $q$-generalized rate function; (iii) the class of strong correlations that we have focused on corresponds to attractors in the sense of the Central Limit Theorem which are $Q$-Gaussian distributions (in principle $1 < Q < 3$), which relevantly differ from (symmetric) L\'evy distributions, with the unique exception of Cauchy-Lorentz distributions (which correspond to $Q = 2$), where they coincide, as well known. In his Comment, Touchette has agreeably discussed point (i), but, unfortunately, points (ii) and (iii) have, as we detail here, visibly escaped to his analysis. Consequently, his conclusion claiming the absence of special connection with $q$-exponentials is unjustified.