Large Deviation Properties Research Articles

Convergence to the Asymptotic Large Deviation Limit.

Large deviation theory offers a powerful and general statistical framework to study the asymptotic dynamical properties of rare events. The application of the formalism to concrete experimental situations is, however, often restricted by finite statistics. Data might not suffice to reach the asymptotic regime or judge whether large deviation estimators converge at all. We here experimentally study the large deviation properties of the stochastic work and heat of a levitated nanoparticle subjected to nonequilibrium feedback control. This setting allows us to determine for each quantity the convergence domain of the large deviation estimators using a criterion that does not require the knowledge of the probability distribution. By extracting both the asymptotic exponential decay and the subexponential prefactors, we demonstrate that singular prefactors significantly restrict the convergence characteristics close to the singularity. Our results provide unique insight into the approach to the asymptotic large deviation limit and underscore the pivotal role of singular prefactors.

Large deviations and conditioning for chaotic non-invertible deterministic maps: analysis via the forward deterministic dynamics and the backward stochastic dynamics

The large deviation properties of trajectory observables for chaotic non-invertible deterministic maps as studied recently by Smith (2022 Phys. Rev. E 106 L042202) and by Gutierrez et al (2023 arXiv:2304.13754) are revisited in order to analyze in detail the similarities and the differences with the case of stochastic Markov chains. More concretely, we focus on the simplest example displaying the two essential properties of local stretching and global folding, namely the doubling map xt+1=2xt[mod1] on the real-space interval x∈[0,1[ that can also be analyzed via the decomposition x=∑l=1+∞σl2l into binary coefficients σl=0,1 . The large deviation properties of trajectory observables can be studied either via deformations of the forward deterministic dynamics or via deformations of the backward stochastic dynamics. Our main conclusions concerning the construction of the corresponding Doob canonical conditioned processes are: (i) non-trivial conditioned dynamics can be constructed only in the backward stochastic perspective where the reweighting of existing transitions is possible, and not in the forward deterministic perspective; (ii) the corresponding conditioned steady state is not smooth on the real-space interval x∈[0,1[ and can be better characterized in the binary space σl=1,2,..,+∞ . As a consequence, the backward stochastic dynamics in the binary space are also the most appropriate framework to analyze higher levels of large deviations, and we obtain the explicit large deviations at level 2 for the probability of the empirical density of long backward trajectories.

Semi-Markov processes in open quantum systems. II. Counting statistics with resetting.

A semi-Markov process method for obtaining general counting statistics for open quantum systems is extended to the scenario of resetting. The simultaneous presence of random resets and wave function collapses means that the quantum jump trajectories are no longer semi-Markov. However, focusing on trajectories and using simple probability formulas, general counting statistics can still be constructed from reset-free statistics. An exact tilted matrix equationis also obtained. The inputs of these methods are the survival distributions and waiting-time density distributions instead of quantum operators. In addition, a continuous-time cloning algorithm is introduced to simulate the large-deviation properties of open quantum systems. Several quantum optics systems are used to demonstrate these results.

Large deviations for the Pearson family of ergodic diffusion processes involving a quadratic diffusion coefficient and a linear force

The Pearson family of ergodic diffusions with a quadratic diffusion coefficient and a linear force is characterized by explicit dynamics of their integer moments and by explicit relaxation of spectral properties towards their steady state. Besides the Ornstein–Uhlenbeck process with a Gaussian steady state, other representative examples of the Pearson family are the square root or the Cox–Ingersoll–Ross process converging towards the gamma distribution, the Jacobi process converging towards the beta distribution, the reciprocal gamma process (corresponding to an exponential functional of the Brownian motion) that converges towards the inverse-gamma distribution, the Fisher–Snedecor process and the Student process. The last three steady states display heavy tails. The goal of the present paper is to analyze the large deviation properties of these various diffusion processes in a unified framework. We first consider level 1 concerning time-averaged observables over a large time window T. We write the first rescaled cumulants for generic observables and identify specific observables whose large deviations can be explicitly computed from the dominant eigenvalue of the appropriate deformed generator. The explicit large deviations at level 2 concerning the time-averaged density are then used to analyze the statistical inference of model parameters from data on a very long stochastic trajectory in order to obtain the explicit rate function for the two inferred parameters of the Pearson linear force.

Large-deviation properties of SIR model incorporating protective measures

We simulate spreads of diseases for the susceptible–infected–recovered (SIR) model on contact networks with a particular focus on incorporated protective measures such as face masks. We consider the small-world network model. By using a large-deviation approach, in particular the Wang–Landau algorithm, we obtained the full probability density function (pdf) of the cumulative number C of infected people over the full range of its support. In this way we are able to reach probabilities as small as 10−50. We obtain distinct characteristics in the pdf such as non-analyticities induced by the onset of the protective measures. Still, the results indicate that the mathematical large-deviation principle also holds for this extended SIR model, meaning that the size-dependence enters in P(C) in a simple fashion and the distribution is determined by the so-called rate function. We observe different phases in the pdf, which we investigate by analyzing the corresponding infection courses, i.e. time series, and and their correlations to the observed values of C.

Quenched and averaged large deviations for random walks in random environments: The impact of disorder

In 2003, Varadhan (Comm. Pure Appl. Math. 56 (2003) 1222–1245) developed a robust method for proving quenched and averaged large deviations for random walks in a uniformly elliptic and i.i.d. environment (RWRE) on Zd. One fundamental question which remained open was to determine when the quenched and averaged large deviation rate functions agree, and when they do not. In this article we show that for RWRE in uniformly elliptic and i.i.d. environment in d≥4, the two rate functions agree on any compact set contained in the interior of their domain which does not contain the origin, provided that the disorder of the environment is sufficiently low. Our result provides a new formulation which encompasses a set of sufficient conditions under which these rate functions agree without assuming that the RWRE is ballistic (see (Probab. Theory Related Fields 149 (2011) 463–491)), satisfies a CLT or even a law of large numbers (Electron. Commun. Probab. 7 (2002)191–197; Ann. Probab. 36 (2008) 728–738). Also, the equality of rate functions is not restricted to neighborhoods around given points, as long as the disorder of the environment is kept low. One of the novelties of our approach is the introduction of an auxiliary random walk in a deterministic environment which is itself ballistic (regardless of the actual RWRE behavior) and whose large deviation properties approximate those of the original RWRE in a robust manner, even if the original RWRE is not ballistic itself.

Joint distribution of two local times for diffusion processes with the application to the construction of various conditioned processes

For a diffusion process X(t) of drift and of diffusion coefficient , we study the joint distribution of the two local times and at positions x = 0 and x = L, as well as the simpler statistics of their sum . Their asymptotic statistics for large time involves two very different cases: (i) when the diffusion process X(t) is transient, the two local times remain finite random variables and we analyze their limiting joint distribution; (ii) when the diffusion process X(t) is recurrent, we describe the large deviations properties of the two intensive local times and and of their intensive sum . These properties are then used to construct various conditioned processes satisfying certain constraints involving the two local times, thereby generalizing our previous work (Mazzolo and Monthus 2022 J. Stat. Mech. 103207) concerning the conditioning with respect to a single local time A(t). In particular for the infinite time horizon , we consider the conditioning towards the finite asymptotic values or , as well as the conditioning towards the intensive values or , that can be compared with the appropriate ‘canonical conditioning’ based on the generating function of the local times in the regime of large deviations. This general construction is then applied to the simplest case where the unconditioned diffusion is the Brownian motion of uniform drift µ.

Stochastic linearized generalized alternating direction method of multipliers: Expected convergence rates and large deviation properties

AbstractAlternating direction method of multipliers (ADMM) receives much attention in the field of optimization and computer science, etc. The generalized ADMM (G-ADMM) proposed by Eckstein and Bertsekas incorporates an acceleration factor and is more efficient than the original ADMM. However, G-ADMM is not applicable in some models where the objective function value (or its gradient) is computationally costly or even impossible to compute. In this paper, we consider the two-block separable convex optimization problem with linear constraints, where only noisy estimations of the gradient of the objective function are accessible. Under this setting, we propose a stochastic linearized generalized ADMM (called SLG-ADMM) where two subproblems are approximated by some linearization strategies. And in theory, we analyze the expected convergence rates and large deviation properties of SLG-ADMM. In particular, we show that the worst-case expected convergence rates of SLG-ADMM are $\mathcal{O}\left( {{N}^{-1/2}}\right)$ and $\mathcal{O}\left({\ln N} \cdot {N}^{-1}\right)$ for solving general convex and strongly convex problems, respectively, where N is the iteration number, similarly hereinafter, and with high probability, SLG-ADMM has $\mathcal{O}\left ( \ln N \cdot N^{-1/2} \right ) $ and $\mathcal{O}\left ( \left ( \ln N \right )^{2} \cdot N^{-1} \right ) $ constraint violation bounds and objective error bounds for general convex and strongly convex problems, respectively.

Nonequilibrium diffusion processes via non-Hermitian electromagnetic quantum mechanics with application to the statistics of entropy production in the Brownian gyrator.

The nonequilibrium Fokker-Planck dynamics in an arbitrary force field f[over ⃗](x[over ⃗]) in dimension N is revisited via the correspondence with the non-Hermitian quantum mechanics in a real scalar potential V(x[over ⃗]) and in a purely imaginary vector potential [-iA[over ⃗](x[over ⃗])] of real amplitude A[over ⃗](x[over ⃗]). The relevant parameters of irreversibility are then the N(N-1)/2 magnetic matrix elements B_{nm}(x[over ⃗])=-B_{mn}(x[over ⃗])=∂_{n}A_{m}(x[over ⃗])-∂_{m}A_{n}(x[over ⃗]), while it is enlightening to explore the corresponding gauge transformations of the vector potential A[over ⃗](x[over ⃗]). This quantum interpretation is even more fruitful to study the statistics of all the time-additive observables of the stochastic trajectories, since their generating functions correspond to the same quantum problem with additional scalar and/or vector potentials. Our main conclusion is that the analysis of their large deviations properties and the construction of the corresponding Doob conditioned processes can be drastically simplified via the choice of an appropriate gauge for each purpose. This general framework is then applied to the special time-additive observables of Ornstein-Uhlenbeck trajectories in dimension N, whose generating functions correspond to quantum propagators involving quadratic scalar potentials and linear vector potentials, i.e., to quantum harmonic oscillators in constant magnetic matrices. As simple illustrative example, we finally focus on the Brownian gyrator in dimension N=2 to compute the large deviations properties of the entropy production of its stochastic trajectories and to construct the corresponding conditioned processes having a given value of the entropy production per unit time.

Large-time behaviour and the second eigenvalue problem for finite-state mean-field interacting particle systems

AbstractThis article examines large-time behaviour of finite-state mean-field interacting particle systems. Our first main result is a sharp estimate (in the exponential scale) of the time required for convergence of the empirical measure process of the N-particle system to its invariant measure; we show that when time is of the order $\exp\{N\Lambda\}$ for a suitable constant $\Lambda > 0$ , the process has mixed well and it is close to its invariant measure. We then obtain large-N asymptotics of the second-largest eigenvalue of the generator associated with the empirical measure process when it is reversible with respect to its invariant measure. We show that its absolute value scales as $\exp\{{-}N\Lambda\}$ . The main tools used in establishing our results are the large deviation properties of the empirical measure process from its large-N limit. As an application of the study of large-time behaviour, we also show convergence of the empirical measure of the system of particles to a global minimum of a certain ‘entropy’ function when particles are added over time in a controlled fashion. The controlled addition of particles is analogous to the cooling schedule associated with the search for a global minimum of a function using the simulated annealing algorithm.

Rare event asymptotics for exploration processes for random graphs

Large deviations for random graph models has been a topic of significant recent research activity. Much work in this area is focused on the class of dense random graph models (number of edges in the graph scale as n2, where n is the number of vertices) where the theory of graphons has emerged as a principal tool in the study of large deviation properties. These tools do not give a good approach to large deviation problems for random graph models in the sparse regime. The aim of this paper is to study an approach for large deviation problems in this regime by establishing large deviation principles (LDP) on suitable path spaces for certain exploration processes of the associated random graph sequence. Exploration processes are an important tool in the study of sparse random graph models and have been used to understand detailed asymptotics of many functionals of sparse random graphs, such as component sizes, surplus, deviations from trees, etc. In the context of rare event asymptotics of interest here, the point of view of exploration process transforms a large deviation analysis of a static random combinatorial structure to the study of a small noise LDP for certain stochastic dynamical systems with jumps. Our work focuses on one particular class of random graph models, namely the configuration model; however, the general approach of using exploration processes for studying large deviation properties of sparse random graph models has broader applicability. The goal is to study asymptotics of probabilities of nontypical behavior in the large network limit. The first key step for this is to establish a LDP for an exploration process associated with the configuration model. A suitable exploration process here turns out to be an infinite-dimensional Markov process with transition probability rates that diminish to zero in certain parts of the state space. Large deviation properties of such Markovian models is challenging due to poor regularity behavior of the associated local rate functions. Our proof of the LDP relies on a representation of the exploration process in terms of a system of stochastic differential equations driven by Poisson random measures and variational formulas for moments of nonnegative functionals of Poisson random measures. Uniqueness results for certain controlled systems of deterministic equations play a key role in the analysis. Next, using the rate function in the LDP for the exploration process we formulate a calculus of variations problem associated with the asymptotics of component degree distributions. The second key ingredient in our study is a careful analysis of the infinite-dimensional Euler–Lagrange equations associated with this calculus of variations problem. Exact solutions of these systems of nonlinear differential equations are identified which then provide explicit formulas for decay rates of probabilities of nontypical component degree distributions and related quantities.

Large deviations of a susceptible-infected-recovered model around the epidemic threshold.

We numerically study the dynamics of the SIR disease model on small-world networks by using a large-deviation approach. This allows us to obtain the probability density function of the total fraction of infected nodes and of the maximum fraction of simultaneously infected nodes down to very small probability densities like 10^{-2500}. We analyze the structure of the disease dynamics and observed three regimes in all probability density functions, which correspond to quick mild, quick extremely severe, and sustained severe dynamical evolutions, respectively. Furthermore, the mathematical rate functions of the densities are investigated. The results indicate that the so-called large-deviation property holds for the SIR model. Finally, we measured correlations with other quantities like the duration of an outbreak or the peak position of the fraction of infections, also in the rare regions which are not accessible by standard simulation techniques.

Bayes Posterior Convergence for Loss Functions via Almost Additive Thermodynamic Formalism.

Statistical inference can be seen as information processing involving input information and output information that updates belief about some unknown parameters. We consider the Bayesian framework for making inferences about dynamical systems from ergodic observations, where the Bayesian procedure is based on the Gibbs posterior inference, a decision process generalization of standard Bayesian inference (see [7, 37]) where the likelihood is replaced by the exponential of a loss function. In the case of direct observation and almost-additive loss functions, we prove an exponential convergence of the a posteriori measures to a limit measure. Our estimates on the Bayes posterior convergence for direct observation are related and extend those in [47] to a context where loss functions are almost-additive. Our approach makes use of non-additive thermodynamic formalism and large deviation properties [39, 40, 57] instead of joinings.

Anomalous dynamical large deviations of local empirical densities and activities in the pure and in the random kinetically constrained East model

The East model is the simplest one-dimensional kinetically-constrained model of $N$ spins with a trivial equilibrium that displays anomalously large spatio-temporal fluctuations, with characteristic "space-time bubbles" in trajectory space, and with a discontinuity at the origin for the first derivative of the scaled cumulant generating function of the total activity. These striking dynamical properties are revisited via the large deviations at various levels for the relevant local empirical densities and activities that only involve two consecutive spins. This framework allows to characterize their anomalous rate functions and to analyze the consequences for all the time-additive observables that can be reconstructed from them, both for the pure and for the random East model. These singularities in dynamical large deviations properties disappear when the hard-constraint of the East model is replaced by the soft constraint.

Large deviations for metastable states of Markov processes with absorbing states with applications to population models in stable or randomly switching environment

The large deviations at level 2.5 are applied to Markov processes with absorbing states in order to obtain the explicit extinction rate of metastable quasi-stationary states in terms of their empirical time-averaged density and of their time-averaged empirical flows over a large time-window T. The standard spectral problem for the slowest relaxation mode can be recovered from the full optimization of the extinction rate over all these empirical observables and the equivalence can be understood via the Doob generator of the process conditioned to survive up to time T. The large deviation properties of any time-additive observable of the Markov trajectory before extinction can be derived from the level 2.5 via the decomposition of the time-additive observable in terms of the empirical density and the empirical flows. This general formalism is described for continuous-time Markov chains, with applications to population birth–death model in a stable or in a switching environment, and for diffusion processes in dimension d.

Inhomogeneous asymmetric exclusion processes between two reservoirs: large deviations for the local empirical observables in the mean-field approximation

For a given inhomogeneous exclusion processes on N sites between two reservoirs, the trajectories probabilities allow to identify the relevant local empirical observables and to obtain the corresponding rate function at level 2.5. In order to close the hierarchy of the empirical dynamics that appear in the stationarity constraints, we consider the simplest approximation, namely the mean-field approximation for the empirical density of two consecutive sites, in direct correspondence with the previously studied mean-field approximation for the steady state. For a given inhomogeneous totally asymmetric model, this mean-field approximation yields the large deviations for the joint distribution of the empirical density profile and of the empirical current around the mean-field steady state; the further explicit contraction over the current allows to obtain the large deviations of the empirical density profile alone. For a given inhomogeneous asymmetric model, the local empirical observables also involve the empirical activities of the links and of the reservoirs; the further explicit contraction over these activities yields the large deviations for the joint distribution of the empirical density profile and of the empirical current. The consequences for the large deviations properties of time-additive space-local observables are also discussed in both cases.

Stability of large complex systems with heterogeneous relaxation dynamics

We study the probability of stability of a large complex system of size N within the framework of a generalized May model, which assumes a linear dynamics of each population size n i (with respect to its equilibrium value): . The a i > 0’s are the intrinsic decay rates, J ij is a real symmetric (N × N) Gaussian random matrix and measures the strength of pairwise interaction between different species. Our goal is to study how inhomogeneities in the intrinsic damping rates a i affect the stability of this dynamical system. As the interaction strength T increases, the system undergoes a phase transition from a stable phase to an unstable phase at a critical value T = T c. We reinterpret the probability of stability in terms of the hitting time of the level b = 0 of an associated Dyson Brownian motion (DBM), starting at the initial position a i and evolving in ‘time’ T. In the large N → ∞ limit, using this DBM picture, we are able to completely characterize T c for arbitrary density μ(a) of the a i ’s. For a specific flat configuration , we obtain an explicit parametric solution for the limiting (as N → ∞) spectral density for arbitrary T and σ. For finite but large N, we also compute the large deviation properties of the probability of stability on the stable side T < T c using a Coulomb gas representation.

Efficiency large deviation function of quantum heat engines

The efficiency of small thermal machines is typically a fluctuating quantity. We here study the efficiency large deviation properties of two exemplary quantum heat engines, the harmonic oscillator and the two-level Otto motors. To this end, we analytically compute their joint characteristic functions for heat and work based on the two-projective-measurement approach. We investigate work–heat correlations within the respective engine cycles and find, for generic scale-invariant quantum heat engines, that work and heat are perfectly anticorrelated for adiabatic driving. In this limit, the effects of thermal as well as quantum fluctuations are suppressed, the large deviation functions are singular and the stochastic efficiency is equal to the macroscopic efficiency.

Inference of Markov models from trajectories via large deviations at level 2.5 with applications to random walks in disordered media

The inference of Markov models from data on stochastic dynamical trajectories over the large time-window T is revisited via the large deviations at level 2.5 for the time-empirical density and the time-empirical flows. The goal is to obtain the large deviations properties for the probability distribution of the inferred Markov parameters in order to characterize their possible fluctuations around the true Markov parameters for large T. The explicit rate functions are given for several settings, namely discrete-time Markov chains, continuous-time Markov jump processes, and diffusion processes in dimension d. Applications to various models of random walks in disordered media are described, where the goal is to infer the quenched disordered variables defining a given disordered sample.

Large deviations in continuous-time random walks.

We discuss large deviation properties of continuous-time random walks (CTRWs) and present a general expression for the large deviation rate in CTRWs in terms of the corresponding rates for the distributions of steps' lengths and waiting times. In the case of Gaussian distribution of steps' lengths the general expression reduces to a sequence of two Legendre transformations applied to the cumulant generating function of waiting times. The discussion of several examples (Bernoulli and Gaussian random walks with exponentially distributed waiting times, Gaussian random walks with one-sided Lévy and Pareto-distributed waiting times) reveals interesting general properties of such large deviations.