Condensation transition in large deviations of self-similar Gaussian processes with stochastic resetting – Archive ouverte HAL

Naftali R. SmithSatya N. Majumdar ¹ Naftali SmithSatya Majumdar ¹

Naftali R. Smith, Satya N. Majumdar, Naftali Smith, Satya Majumdar. Condensation transition in large deviations of self-similar Gaussian processes with stochastic resetting. Journal of Statistical Mechanics: Theory and Experiment, IOP Publishing, 2022, 2022 (5), pp.053212. ⟨10.1088/1742-5468/ac6f04⟩. ⟨hal-03831987⟩

Abstract We study the fluctuations of the area A ( t ) = ∫ 0 t x ( τ ) d τ under a self-similar Gaussian process x ( τ ) with Hurst exponent H > 0 (e.g., standard or fractional Brownian motion, or the random acceleration process) that stochastically resets to the origin at rate r . Typical fluctuations of A ( t ) scale as ∼ t for large t and on this scale the distribution is Gaussian, as one would expect from the central limit theorem. Here our main focus is on atypically large fluctuations of A ( t ). In the long-time limit t → ∞, we find that the full distribution of the area takes the form P r A | t ∼ exp − t α Φ A / t β with anomalous exponents α = 1/(2 H + 2) and β = (2 H + 3)/(4 H + 4) in the regime of moderately large fluctuations, and a different anomalous scaling form P r A | t ∼ exp − t Ψ A / t 2 H + 3 / 2 in the regime of very large fluctuations. The associated rate functions Φ( y ) and Ψ( w ) depend on H and are found exactly. Remarkably, Φ( y ) has a singularity that we interpret as a first-order dynamical condensation transition, while Ψ( w ) exhibits a second-order dynamical phase transition above which the number of resetting events ceases to be extensive. The parabolic behavior of Φ( y ) around the origin y = 0 correctly describes the typical, Gaussian fluctuations of A ( t ). Despite these anomalous scalings, we find that all of the cumulants of the distribution P r A | t grow linearly in time, ⟨ A n ⟩ c ≈ c n t , in the long-time limit. For the case of reset Brownian motion (corresponding to H = 1/2), we develop a recursive scheme to calculate the coefficients c n exactly and use it to calculate the first six nonvanishing cumulants.

• 1. LPTMS – Laboratoire de Physique Théorique et Modèles Statistiques