Asymmetric scaling in large deviations for rare values bigger or smaller than the typical value

Abstract

In various disordered systems or non-equilibrium dynamical models, the large deviations of some observables have been found to display different scalings for rare values bigger or smaller than the typical value. In the present paper, we revisit the simpler observables based on independent random variables, namely the empirical maximum, the empirical average, the empirical non-integer moments or other additive empirical observables, in order to describe the cases where asymmetric large deviations already occur. The unifying starting point to analyze the large deviations of these various empirical observables is given by the Sanov theorem for the large deviations of the empirical histogram : the rate function corresponds to the relative entropy with respect to the true probability distribution and it can be optimized in the presence of the appropriate constraints. Finally, the physical meaning of large deviations rate functions is discussed from the renormalization perspective.I.

Real-space renormalization for disordered systems at the level of large deviations

“…In this appendix, we have considered as in [87] the problem of the tail x → +∞ of the empirical average of a finite number A of independent variables, while the standard large deviations problem for the empirical average focuses instead on a large number A → +∞ of independent variables, while x remains finite. The two problems are thus clearly dierent, but they nevertheless display some similarities as discussed in detail in [87], and the two democratic/monocratic behaviors have also been much studied in the large deviation regime [30][31][32][33][34][35].…”

“…The typical fluctuations are classified in terms of the Gaussian distribution of the central limit theorem (see [27][28][29] for the renormalization point of view) and in terms of the Lévy stable laws (when the variance does not exist). While the standard theory for the large deviations of the empirical average focuses on the case of symmetric large deviations [11,13], the case of asymmetric large deviations (with dierent scalings for rare values bigger or smaller than the typical value) have also attracted a lot of attention recently [30][31][32][33][34][35]. As recalled in appendix, the tails properties of the empirical average of equation (3) strongly depend on the tail exponents α ± of the initial condition of equation (2) with completely dierent regimes associated to compressed exponentials α ± > 1, stretched exponentials 0 < α ± < 1 and simple exponentials α ± = 1.…”

“…Mech. (2020) 013301 properties of the empirical maximum have been found to be asymmetric [35,46,47], as a consequence of the following obvious asymmetry: an 'anomalously good' maximum requires only one anomalously good variable, while an 'anomalously bad' maximum requires that all variables are anomalously bad. This simple argument allows to understand why the large deviations will be also completely dierent for the two tails x → ±∞ in the more general problem of equation (1).…”

“…The typical fluctuations are classified in terms of the three universality classes Gumbel-Fréchet-Weibull [36,37], with many applications in various physics domains (see the reviews [38][39][40] and references therein) and have been much studied from the renormalization perspective [41][42][43][44][45]. The large deviations properties of the empirical maximum have been found to be asymmetric [35,46,47], as a consequence of the following obvious asymmetry : an 'anomalously good' maximum requires only one anomalously good variable, while an 'anomalously bad' maximum requires that all variables are anomalously bad. This simple argument allows to understand why the large deviations will be also completely different for the two tails x → ±∞ in the more general problem of Eq.…”

“…Since the symmetric and asymmetric large deviations properties of these two special cases (i) and (ii) have been revisited in great detail recently in the companion paper [35], we will focus here on the non-degenerate cases (A > 1, B > 1) where there is really an interplay between the maximum and the sum operations. The paper is organized as follows.…”