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Dynamical large deviations, self-similarity, and criticality

Vittoria Sposini*, Francesco Arceri†, Enzo Orlandini‡, Flavio Seno§, and Fulvio Baldovin∥

  • Dipartimento di Fisica e Astronomia ‘G. Galilei’ (DFA), Sezione INFN, Università di Padova, Via Marzolo 8, 35131 Padova (PD), Italy

  • *Contact author: vittoria.sposini@unipd.it
  • †Contact author: francesco.arceri@pd.infn.it
  • ‡Contact author: enzo.orlandini@unipd.it
  • §Contact author: flavio.seno@unipd.it
  • ∥Contact author: fulvio.baldovin@unipd.it
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Phys. Rev. Research 8, 043136 – Published 6 November, 2026

DOI: https://doi.org/10.1103/bx6m-13rm

Abstract

Large-deviation theory describes the statistics of rare fluctuations, while self-similarity captures scale-invariant behavior. Both are central pillars of statistical physics, yet the extent of their overlap is not well understood. Here, we ask whether a stochastic process can simultaneously satisfy both. We show that, under a precise set of conditions—exact self-similarity governed by one Hurst exponent and a scaled cumulant-generating function that is analytic around the origin—compatibility holds only if the scaling distribution is Gaussian. We illustrate this result with a broad class of subordinated stochastic processes, including continuous-time random walks, diffusing-diffusivity models, and polymers in the grand-canonical ensemble. In non-Gaussian cases, a characteristic timescale separates regular large-deviation and self-similar regimes. At criticality, the divergence of the correlation time and the emergence of a singular cumulant generating function make a regular large-deviation description inappropriate and lead to a strong enhancement of extreme dynamical fluctuations.

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Phys. Rev. Research 8, 043136– Published 6 November, 2026
  • Received 30 January 2026
  • Accepted 9 October 2026
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