Subleading-order theory for condensation transitions in large deviations of sums of independent and identically distributed random variables

Abstract

We study the full distribution $P_N\left(A\right)$ of sums $A={\sum }_{i=1}^N$ where $x_1,\dots ,x_N$ are $N\gg 1$ independent and identically distributed random variables each sampled from a given distribution $p(x)$ with a subexponential $x\to \infty$ tail. We consider two particular cases: (I) the one-sided stretched exponential distribution $p(x)\propto e^{-x^{\alpha }}$ where $0<x<\infty$, (II) the two-sided stretched exponential distribution $p(x)\propto e^{-|x{|}^{\alpha }}$ where $-\infty <x<\infty$. We assume $0<\alpha <1$ (in both cases). As follows immediately from known theorems, for both cases (i) typical fluctuations of $\Delta A=A-⟨A⟩$ are described by the central-limit theorem, (ii) the tail $A\to \infty$ is described by the big-jump principle $P_N\left(A\right)\simeq Np\left(A\right)$, and (iii) in between these two regimes there is a nontrivial intermediate regime which displays anomalous scaling $P_N\left(A\right)\sim e^{-N^{\beta }f(\Delta A/N^{\gamma })}$ with anomalous exponents $\beta ,\gamma \in (0,1)$ and large-deviation function $f(y)$ that are all exactly known. In practice, although these theoretical predictions of $P_N\left(A\right)$ work very well in regimes (i) and (ii), they often perform quite poorly in the intermediate regime (ii), with errors of several orders of magnitude for $N$ as large as $10^4$. We calculate subleading order corrections to the theoretical predictions in the intermediate regime. We find that for $0<\alpha <{\alpha }_c$, these corrections scale as power laws in $N$, while for ${\alpha }_c<\alpha <1$ they scale as stretched exponentials, where the threshold value is ${\alpha }_c=1/2$ in case (I) and ${\alpha }_c=2/3$ in case (II). This difference between the two cases is a result of the mirror symmetry $p(x)=p(-x)$ which holds only in the latter case.

19 pages, 7 figures

References in corpus (44)

• The large deviation approach to statistical mechanics
• Nonequilibrium Statistical Mechanics of the Zero-Range Process and Related Models
• Introduction to dynamical large deviations of Markov processes
• Nonequilibrium Phase Transitions in Models of Aggregation, Adsorption, and Dissociation
• The Nature of the Condensate in Mass Transport Models
• Canonical Analysis of Condensation in Factorised Steady State
• Generalised extreme value statistics and sum of correlated variables
• Statistical distribution of quantum entanglement for a random bipartite state
• Large deviations for random walks under subexponentiality: the big-jump domain
• Phase Transitions in the Distribution of Bipartite Entanglement of a Random Pure State
• The Single Big Jump Principle in Physical Modelling
• A First-Order Dynamical Transition in the displacement distribution of a Driven Run-and-Tumble Particle
• Properties of additive functionals of Brownian motion with resetting
• Lévy walks and scaling in quenched disordered media
• Anomalous scaling of dynamical large deviations
• Condensation transition in the late-time position of a Run-and-Tumble particle
• Finite size scaling of the balls in boxes model
• Localization transition in the Discrete Non-Linear Schrödinger Equation: ensembles inequivalence and negative temperatures
• Rare events in generalized Lévy Walks and the Big Jump principle
• Dynamical large deviations of reflected diffusions
• Transport in disordered systems: the single big jump approach
• Condensation of Fluctuations in and out of Equilibrium
• Rare events and scaling properties in field-induced anomalous dynamics
• Anomalous scaling and first-order dynamical phase transition in large deviations of the Ornstein-Uhlenbeck process
• Condensation transition and ensemble inequivalence in the Discrete Nonlinear Schrödinger Equation
• Anomalous scaling of dynamical large deviations of stationary Gaussian processes
• Large deviations, condensation, and giant response in a statistical system
• Superdiffusion and Transport in 2d-systems with Lévy Like Quenched Disorder
• Large deviations in chaotic systems: exact results and dynamical phase transition
• Dynamic fluctuations in unfrustrated systems: random walks, scalar fields and the Kosterlitz-Thouless phase
• Real-space Condensation in Stochastic Mass Transport Models
• Fast rare events in exit times distributions of jump processes
• Anomalous scalings of fluctuations of the area swept by a Brownian particle trapped in a $|x|$ potential
• Large deviations and phase transitions in spectral linear statistics of Gaussian random matrices
• Finding the effective dynamics to make rare events typical in chaotic maps
• Large deviations and conditioning for chaotic non-invertible deterministic maps: analysis via the forward deterministic dynamics and the backward stochastic dynamics
• Rare Events and Single Big Jump Effects in Ornstein-Uhlenbeck Processes
• Rare Events in Extreme Value Statistics of Jump Processes with Power Tails
• Dynamical large deviations of the fractional Ornstein-Uhlenbeck process
• Thermodynamics of chaotic relaxation processes
• Explicit dynamical properties of the Pelikan random map in the chaotic region and at the intermittent critical point towards the non-chaotic region
• Data-driven analysis of annual rain distributions
• Optimal finite-differences discretization for the diffusion equation from the perspective of large-deviation theory
• Heavy tailed large deviations for time averages of diffusions: the Ornstein-Uhlenbeck case

Cited by in corpus (1)

• Beyond the Big Jump: A Perturbative Approach to Stretched-Exponential Processes