SciPost Phys. 19, 001 (2025) ·
published 1 July 2025
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We discuss the breakdown of the Parisi-Sourlas supersymmetry (SUSY) and of the dimensional-reduction (DR) property in the random field Ising and O($N$) models as a function of space dimension $d$ and/or number of components $N$. The functional renormalization group (FRG) predicts that this takes place below a critical line $d_\mathrm{DR}(N)$. We revisit the perturbative FRG results for the RFO($N$)M in $d=4+\epsilon$ and carry out a more comprehensive investigation of the nonperturbative FRG approximation for the RFIM. In light of this FRG description, we discuss the perturbative results in $\epsilon=6-d$ recently derived for the RFIM by Kaviraj, Rychkov, and Trevisani. We stress in particular that the disappearance of the SUSY/DR fixed point below $d_\mathrm{DR}$ arises as a consequence of the nonlinearity of the FRG equations and cannot be found via the perturbative expansion in $\epsilon=6-d$ (nor in $1/N$). We also provide an error bar on the value of the critical dimension $d_\mathrm{DR}$ for the RFIM, which we find around $5.11±0.09$, by studying several successive orders of the nonperturbative FRG approximation scheme.
SciPost Phys. 18, 119 (2025) ·
published 2 April 2025
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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 Cramér's series to strongly correlated variables. Our findings highlight the universal and nonuniversal aspects of rare event statistics. We also discuss the ubiquity of this power-law corrections to the leading compressed-exponential decay in these tails in and out-of-equilibrium.
