Large Deviation Regime Research Articles

Large deviations for sticky Brownian motions

We consider n-point sticky Brownian motions: a family of n diffusions that evolve as independent Brownian motions when they are apart, and interact locally so that the set of coincidence times has positive Lebesgue measure with positive probability. These diffusions can also be seen as n random motions in a random environment whose distribution is given by so-called stochastic flows of kernels. For a specific type of sticky interaction, we prove exact formulas characterizing the stochastic flow and show that in the large deviations regime, the random fluctuations of these stochastic flows are Tracy-Widom GUE distributed. An equivalent formulation of this result states that the extremal particle among n sticky Brownian motions has Tracy-Widom distributed fluctuations in the large n and large time limit. These results are proved by viewing sticky Brownian motions as a (previously known) limit of the exactly solvable beta random walk in random environment.

Heavy-tailed random walks, buffered queues and hidden large deviations

It is well-known that large deviations of random walks driven by independent and identically distributed heavy-tailed random variables are governed by the so-called principle of one large jump. We note that further subtleties hold for such random walks in the large deviations scale which we call hidden large deviations. Our results are illustrated using two examples. First, we apply this idea in the context of queueing processes with heavy-tailed service times and study approximations of probabilities of severe congestion times for (buffered) queues. We exhibit our techniques by using limit measures from different large deviation regimes to provide a unified estimate of rare event probabilities in a simulated queue. Furthermore, we use our result to provide probability estimates of rare events governed by more than one jump in case the innovations of a random walk have infinite mean.

Geometrical optics of constrained Brownian motion: three short stories

The optimal fluctuation method—essentially geometrical optics—gives a deep insight into large deviations of Brownian motion. Here we illustrate this point by telling three short stories about Brownian motions, ‘pushed’ into a large-deviation regime by constraints. In story 1 we compute the short-time large deviation function (LDF) of the winding angle of a Brownian particle wandering around a reflecting disk in the plane. Story 2 addresses a stretched Brownian motion above absorbing obstacles in the plane. We compute the short-time LDF of the position of the surviving Brownian particle at an intermediate point. Story 3 deals with survival of a Brownian particle in 1 + 1 dimension against absorption by a wall which advances according to a power law , where . We also calculate the LDF of the particle position at an earlier time, conditional on the survival by a later time. In all three stories we uncover singularities of the LDFs which have a simple geometric origin and can be interpreted as dynamical phase transitions. We also use the small-deviation limit of the geometrical optics to reconstruct the distribution of typical fluctuations. We argue that, in stories 2 and 3, this is the Ferrari–Spohn distribution.

Delocalization of Polymers in Lower Tail Large Deviation

Directed last passage percolation models on the plane, where one studies the weight as well as the geometry of optimizing paths (called polymers) in a field of i.i.d. weights, are paradigm examples of models in KPZ universality class. In this article, we consider the large deviation regime, i.e., when the polymer has a much smaller (lower tail) or larger (upper tail) weight than typical. Precise asymptotics of large deviation probabilities have been obtained in a handful of the so-called exactly solvable scenarios, including the Exponential (Johansson in Commun Math Phys 209(2):437–476, 2000) and Poissonian (Deuschel and Zeitouni in Comb Probab Comput 8(03):247–263, 1999; Seppalainen in Probab Theory Relat Fields 112(2):221–244, 1998) cases. How the geometry of the optimizing paths change under such a large deviation event was considered in Deuschel and Zeitouni (1999) where it was shown that the paths [from (0, 0) to (n, n), say] remain concentrated around the straight line joining the end points in the upper tail large deviation regime, but the corresponding question in the lower tail was left open. We establish a contrasting behaviour in the lower tail large deviation regime, showing that conditioned on the latter, in both the models, the optimizing paths are not concentrated around any deterministic curve. Our argument does not use any ingredient from integrable probability, and hence can be extended to other planar last passage percolation models under fairly mild conditions; and also to other non-integrable settings such as last passage percolation in higher dimensions.

Intermediate deviation regime for the full eigenvalue statistics in the complex Ginibre ensemble.

We study the Ginibre ensemble of N×N complex random matrices and compute exactly, for any finite N, the full distribution as well as all the cumulants of the number N_{r} of eigenvalues within a disk of radius r centered at the origin. In the limit of large N, when the average density of eigenvalues becomes uniform over the unit disk, we show that for 0<r<1 the fluctuations of N_{r} around its mean value 〈N_{r}〉≈Nr^{2} display three different regimes: (i) a typical Gaussian regime where the fluctuations are of order O(N^{1/4}), (ii) an intermediate regime where N_{r}-〈N_{r}〉=O(sqrt[N]), and (iii) a large deviation regime where N_{r}-〈N_{r}〉=O(N). This intermediate behavior (ii) had been overlooked in previous studies and we show here that it ensures a smooth matching between the typical and the large deviation regimes. In addition, we demonstrate that this intermediate regime controls all the (centered) cumulants of N_{r}, which are all of order O(sqrt[N]). We show that the intermediate deviation function that describes these intermediate fluctuations can be computed explicitly and we demonstrate that it is universal, i.e., it holds for a large class of complex random matrices. Our analytical results are corroborated by precise "importance sampling" Monte Carlo simulations.

Condensation for random variables conditioned by the value of their sum

We revisit the problem of condensation for independent, identically distributed random variables with a power-law tail, conditioned by the value of their sum. For large values of the sum, and for a large number of summands, a condensation transition occurs where the largest summand accommodates the excess difference between the value of the sum and its mean. This simple scenario of condensation underlies a number of studies in statistical physics, such as, e.g. in random allocation and urn models, random maps, zero-range processes and mass transport models. Much of the effort here is devoted to presenting the subject in simple terms, reproducing known results and adding some new ones. In particular we address the question of the quantitative comparison between asymptotic estimates and exact finite-size results. Simply stated, one would like to know how accurate are the asymptotic estimates of the observables of interest, compared to their exact finite-size counterparts, to the extent that they are known. This comparison, illustrated on the particular exemple of a distribution with Lévy index equal to 3/2, demonstrates the role of the contributions of the dip and large deviation regimes. Except for the last section devoted to a brief review of extremal statistics, the presentation is self-contained and uses simple analytical methods.

Efficient Rare-Event Simulation for Multiple Jump Events in Regularly Varying Random Walks and Compound Poisson Processes

We propose a class of strongly efficient rare-event simulation estimators for random walks and compound Poisson processes with a regularly varying increment/jump-size distribution in a general large deviations regime. Our estimator is based on an importance sampling strategy that hinges on a recently established heavy-tailed sample-path large deviations result. The new estimators are straightforward to implement and can be used to systematically evaluate the probability of a wide range of rare events with bounded relative error. They are “universal” in the sense that a single importance sampling scheme applies to a very general class of rare events that arise in heavy-tailed systems. In particular, our estimators can deal with rare events that are caused by multiple big jumps (therefore, beyond the usual principle of a single big jump) as well as multidimensional processes such as the buffer content process of a queueing network. We illustrate the versatility of our approach with several applications that arise in the context of mathematical finance, actuarial science, and queueing theory.

Anomalous one-dimensional fluctuations of a simple two-dimensional random walk in a large-deviation regime.

The following question is the subject of our work: could a two-dimensional (2D) random path pushed by some constraints to an improbable "large-deviation regime" possess extreme statistics with one-dimensional (1D) Kardar-Parisi-Zhang (KPZ) fluctuations? The answer is positive, though nonuniversal, since the fluctuations depend on the underlying geometry. We consider in detail two examples of 2D systems for which imposed external constraints force the underlying stationary stochastic process to stay in an atypical regime with anomalous statistics. The first example deals with the fluctuations of a stretched 2D random walk above a semicircle or a triangle. In the second example we consider a 2D biased random walk along a channel with forbidden voids of circular and triangular shapes. In both cases we are interested in the dependence of a typical span 〈d(t)〉∼t^{γ} of the trajectory of t steps above the top of the semicircle or the triangle. We show that γ=1/3, i.e., 〈d(t)〉 shares the KPZ statistics for the semicircle, while γ=0 for the triangle. We propose heuristic derivations of scaling exponents γ for different geometries, justify them by explicit analytic computations, and compare with numeric simulations. For practical purposes, our results demonstrate that the geometry of voids in a channel might have a crucial impact on the width of the boundary layer and, thus, on the heat transfer in the channel.

Neyman–Pearson Test for Zero-Rate Multiterminal Hypothesis Testing

The problem of zero-rate multiterminal hypothesis testing is revisited from the perspective of information-spectrum approach and finite blocklength analysis. A Neyman–Pearson-like test is proposed and its non-asymptotic performance is clarified, for a short block length, it is numerically determined that the proposed test is superior to the previously reported Hoeffding-like test proposed by Han–Kobayashi. For a large deviation regime, it is shown that our proposed test achieves an optimal trade-off between the type I and type II exponents presented by Han–Kobayashi. Among the class of symmetric (type-based) testing schemes, when the type I error probability is non-vanishing, the proposed test is optimal up to the second-order term of the type II error exponent; the latter term is characterized in terms of the variance of the projected relative entropy density. The information geometry method plays an important role in the analysis as well as the construction of the test.

Extremes of 2d Coulomb gas: universal intermediate deviation regime

In this paper, we study the extreme statistics in the complex Ginibre ensemble of random matrices with complex Gaussian entries, but with no other symmetries. All the N eigenvalues are complex random variables and their joint distribution can be interpreted as a 2d Coulomb gas with a logarithmic repulsion between any pair of particles and in presence of a confining harmonic potential . We study the statistics of the eigenvalue with the largest modulus in the complex plane. The typical and large fluctuations of around its mean had been studied before, and they match smoothly to the right of the mean. However, it remained a puzzle to understand why the large and typical fluctuations to the left of the mean did not match. In this paper, we show that there is indeed an intermediate fluctuation regime that interpolates smoothly between the large and the typical fluctuations to the left of the mean. Moreover, we compute explicitly this ‘intermediate deviation function’ (IDF) and show that it is universal, i.e. independent of the confining potential as long as it is spherically symmetric and increases faster than for large r with an unbounded support. If the confining potential has a finite support, i.e. becomes infinite beyond a finite radius, we show via explicit computation that the corresponding IDF is different. Interestingly, in the borderline case where the confining potential grows very slowly as for with an unbounded support, the intermediate regime disappears and there is a smooth matching between the central part and the left large deviation regime.

Scaling Limits for Infinite-server Systems in a Random Environment

This paper studies the effect of an overdispersed arrival process on the performance of an infinite-server system. In our setup, a random environment is modeled by drawing an arrival rate Λ from a given distribution every Δ time units, yielding an i.i.d. sequence of arrival rates Λ1,Λ2, …. Applying a martingale central limit theorem, we obtain a functional central limit theorem for the scaled queue length process. We proceed to large deviations and derive the logarithmic asymptotics of the queue length’s tail probabilities. As it turns out, in a rapidly changing environment (i.e., Δ is small relative to Λ) the overdispersion of the arrival process hardly affects system behavior, whereas in a slowly changing random environment it is fundamentally different; this general finding applies to both the central limit and the large deviations regime. We extend our results to the setting where each arrival creates a job in multiple infinite-server queues.

Edge statistics for a class of repulsive particle systems

We study a class of interacting particle systems on $$\mathbb {R}$$ which was recently investigated by Gotze and Venker (Ann Probab 42(6):2207–2242, 2014. doi: 10.1214/13-AOP844 ). These ensembles generalize eigenvalue ensembles of Hermitian random matrices by allowing different interactions between particles. Although these ensembles are not known to be determinantal one can use the stochastic linearization method of Gotze and Venker (Ann Probab 42(6):2207–2242, 2014. doi: 10.1214/13-AOP844 ) to represent them as averages of determinantal ones. Our results describe the transition between universal behavior in the regime of the Tracy–Widom law and non-universal behavior for large deviations of the rightmost particle. Moreover, a detailed analysis of the transition that occurs in the regime of moderate deviations, is provided. We also compare our results with the corresponding ones obtained recently for determinantal ensembles (Eichelsbacher et al. in Symmetry Integr Geom Methods Appl 12:18, 2016. doi: 10.3842/SIGMA.2016.093 ; Schuler in Moderate, large and superlarge deviations for extremal eigenvalues of unitarily invariant ensembles. Ph.D. thesis, University of Bayreuth, 2015). In particular, we discuss how the averaging effects the leading order behavior in the regime of large deviations. In the analysis of the averaging procedure we use detailed asymptotic information on the behavior of Christoffel–Darboux kernels that is uniform for perturbative families of weights. Such results have been provided by K. Schubert, K. Schuler and the authors in Kriecherbauer et al. (Markov Process Relat Fields 21(3, part 2):639–694, 2015).

Scaling limits for infinite-server systems in a random environment

This paper studies the effect of an overdispersed arrival process on the performance of an infinite-server system. In our setup, a random environment is modeled by drawing an arrival rate $\Lambda$ from a given distribution every $\Delta$ time units, yielding an i.i.d. sequence of arrival rates $\Lambda_1,\Lambda_2, \ldots$. Applying a martingale central limit theorem, we obtain a functional central limit theorem for the scaled queue length process. We proceed to large deviations and derive the logarithmic asymptotics of the queue length's tail probabilities. As it turns out, in a rapidly changing environment (i.e., $\Delta$ is small relative to $\Lambda$) the overdispersion of the arrival process hardly affects system behavior, whereas in a slowly changing random environment it is fundamentally different; this general finding applies to both the central limit and the large deviations regime. We extend our results to the setting where each arrival creates a job in multiple infinite-server queues.

Exact solution for a random walk in a time-dependent 1D random environment: the point-to-point Beta polymer

We consider the Beta polymer, an exactly solvable model of directed polymer on the square lattice, introduced by Barraquand and Corwin (BC) (2016 Probab. Theory Relat. Fields 1–16). We study the statistical properties of its point to point partition sum. The problem is equivalent to a model of a random walk in a time-dependent (and in general biased) 1D random environment. In this formulation, we study the sample to sample fluctuations of the transition probability distribution function (PDF) of the random walk. Using the Bethe ansatz we obtain exact formulas for the integer moments, and Fredholm determinant formulas for the Laplace transform of the directed polymer partition sum/random walk transition probability. The asymptotic analysis of these formulas at large time t is performed both (i) in a diffusive vicinity, , of the optimal direction (in space-time) chosen by the random walk, where the fluctuations of the PDF are found to be Gamma distributed; (ii) in the large deviations regime, , of the random walk, where the fluctuations of the logarithm of the PDF are found to grow with time as t1/3 and to be distributed according to the Tracy–Widom GUE distribution. Our exact results complement those of BC for the cumulative distribution function of the random walk in regime (ii), and in regime (i) they unveil a novel fluctuation behavior. We also discuss the crossover regime between (i) and (ii), identified as . Our results are confronted to extensive numerical simulations of the model.

Kardar–Parisi–Zhang Equation and Large Deviations for Random Walks in Weak Random Environments

We consider the transition probabilities for random walks in $1+1$ dimensional space-time random environments (RWRE). For critically tuned weak disorder we prove a sharp large deviation result: after appropriate rescaling, the transition probabilities for the RWRE evaluated in the large deviation regime, converge to the solution to the stochastic heat equation (SHE) with multiplicative noise (the logarithm of which is the KPZ equation). We apply this to the exactly solvable Beta RWRE and additionally present a formal derivation of the convergence of certain moment formulas for that model to those for the SHE.

Refined large deviations asymptotics for Markov-modulated infinite-server systems

Many networking-related settings can be modeled by Markov-modulated infinite-server systems. In such models, the customers’ arrival rates and service rates are modulated by a Markovian background process; additionally, there are infinitely many servers (and consequently the resulting model is often used as a proxy for the corresponding many-server model). The Markov-modulated infinite-server model hardly allows any explicit analysis, apart from results in terms of systems of (ordinary or partial) differential equations for the underlying probability generating functions, and recursions to obtain all moments. As a consequence, recent research efforts have pursued an asymptotic analysis in various limiting regimes, notably the central-limit regime (describing fluctuations around the average behavior) and the large-deviations regime (focusing on rare events). Many of these results use the property that the number of customers in the system obeys a Poisson distribution with a random parameter. The objective of this paper is to develop techniques to accurately approximate tail probabilities in the large-deviations regime. We consider the scaling in which the arrival rates are inflated by a factor N, and we are interested in the probability that the number of customers exceeds a given level Na. Where earlier contributions focused on so-called logarithmic asymptotics of this exceedance probability (which are inherently imprecise), the present paper improves upon those results in that exact asymptotics are established. These are found in two steps: first the distribution of the random parameter of the Poisson distribution is characterized, and then this knowledge is used to identify the exact asymptotics. The paper is concluded by a set of numerical experiments, in which the accuracy of the asymptotic results is assessed.

Uniform Random Number Generation From Markov Chains: Non-Asymptotic and Asymptotic Analyses

In this paper, we derive non-asymptotic achievability and converse bounds on the random number generation with/without side-information. Our bounds are efficiently computable in the sense that the computational complexity does not depend on the block length. We also characterize the asymptotic behaviors of the large deviation regime and the moderate deviation regime by using our bounds, which implies that our bounds are asymptotically tight in those regimes. We also show the second order rates of those problems, and derive single letter forms of the variances characterizing the second order rates. Further, we address the equivocation rates for these problems.

Stochastic efficiency for effusion as a thermal engine

The stochastic efficiency of effusion as a thermal engine is investigated within the framework of stochastic thermodynamics. Explicit results are obtained for the probability distribution of the efficiency both at finite times and in the asymptotic regime of large deviations. The universal features, derived in Verley et al. (Nat. Commun., 5 (2014) 4721), are reproduced. The effusion engine is a good candidate for both the numerical and experimental verification of these predictions.

Typical Pure States and Nonequilibrium Processes in Quantum Many-Body Systems

We show that it is possible to calculate equilibrium expectation values of many-body correlated quantities such as the characteristic functions and probability distributions with the use of only a single typical pure state. It also means that we can apply the pure state approach to Heisenberg operators and their spectral fluctuation, and hence to nonequilibrium processes starting from equilibrium. In particular, we can accurately analyze the full statistics of entropy production in nonequilibrium mesoscopic systems. In this way, we can access the full information on higher-order fluctuations in the large deviation regime far from equilibrium.

On the concentration of large deviations for fat tailed distributions, with application to financial data

Large deviations for fat tailed distributions, i.e. those that decay slower than exponential, are not only relatively likely, but they also occur in a rather peculiar way where a finite fraction of the whole sample deviation is concentrated on a single variable. The regime of large deviations is separated from the regime of typical fluctuations by a phase transition where the symmetry between the points in the sample is spontaneously broken. For stochastic processes with a fat tailed microscopic noise, this implies that, while typical realizations are well described by a diffusion process with continuous sample paths, large deviation paths are typically discontinuous. For eigenvalues of random matrices with fat tailed distributed elements, a large deviation where the trace of the matrix is anomalously large concentrates on just a single eigenvalue, whereas in the thin tailed world the large deviation affects the whole distribution.These results find a natural application to finance. Since the price dynamics of financial stocks are characterized by fat tailed increments, large fluctuations in stock prices are expected to be realized by discrete jumps. Interestingly, we find that large excursions of prices are more likely realized by continuous drifts rather than by discontinuous jumps. Indeed, auto correlations suppress the concentration of large deviations. Financial covariance matrices also exhibit an anomalously large eigenvalue, the market mode, as compared to the prediction of random matrix theory. We show that this is explained by a large deviation with excess covariance rather than by one with excess volatility.