Large Deviation Regime Research Articles

Dispersion in the large-deviation regime. Part 2. Cellular flow at large Péclet number

AbstractA standard model for the study of scalar dispersion through the combined effect of advection and molecular diffusion is a two-dimensional periodic flow with closed streamlines inside periodic cells. Over long time scales, the dispersion of a scalar released in this flow can be characterized by an effective diffusivity that is a factor$\mathit{Pe}^{1/2}$larger than molecular diffusivity when the Péclet number$\mathit{Pe}$is large. Here we provide a more complete description of dispersion in this regime by applying the large-deviation theory developed in Part 1 of this paper. Specifically, we derive approximations to the rate function governing the scalar concentration at large time$t$by carrying out an asymptotic analysis of the relevant family of eigenvalue problems. We identify two asymptotic regimes and, for each, make predictions for the rate function and spatial structure of the scalar. Regime I applies to distances$|\boldsymbol {x}|$from the scalar release point that satisfy$|\boldsymbol {x}|= O(\mathit{Pe}^{1/4} t)$. The concentration in this regime is isotropic at large scales, is uniform along streamlines within each cell, and varies rapidly in boundary layers surrounding the separatrices between adjacent cells. The results of homogenization theory, yielding the$O(\mathit{Pe}^{1/2})$effective diffusivity, are recovered from our analysis in the limit$|\boldsymbol {x}|\ll \mathit{Pe}^{1/4} t$. Regime II applies when$|\boldsymbol {x}|=O(\mathit{Pe}\, t/{\rm log}\, \mathit{Pe})$and is characterized by an anisotropic concentration distribution that is localized around the separatrices. A novel feature of this regime is the crucial role played by the dynamics near the hyperbolic stagnation points. A consequence is that in part of the regime the dispersion can be interpreted as resulting from a random walk on the lattice of stagnation points. The two regimes overlap so that our asymptotic results describe the scalar concentration over a large range of distances$|\boldsymbol {x}|$. They are verified against numerical solutions of the family of eigenvalue problems yielding the rate function.

Dispersion in the large-deviation regime. Part 1: shear flows and periodic flows

AbstractThe dispersion of a passive scalar in a fluid through the combined action of advection and molecular diffusion is often described as a diffusive process, with an effective diffusivity that is enhanced compared with the molecular value. However, this description fails to capture the tails of the scalar concentration distribution in initial-value problems. To remedy this, we develop a large-deviation theory of scalar dispersion that provides an approximation to the scalar concentration valid at much larger distances away from the centre of mass, specifically distances that are$O(t)$rather than$O(t^{1/2})$, where$t \gg 1$is the time from the scalar release. The theory centres on the calculation of a rate function characterizing the large-time form of the scalar concentration. This function is deduced from the solution of a one-parameter family of eigenvalue problems which we derive using two alternative approaches, one asymptotic, the other probabilistic. We emphasize the connection between the large-deviation theory and the homogenization theory that is often used to compute effective diffusivities: a perturbative solution of the eigenvalue problems in the appropriate limit reduces at leading order to the cell problem of homogenization theory. We consider two classes of flows in some detail: shear flows and periodic flows with closed streamlines (cellular flows). In both cases, large deviation generalizes classical results on effective diffusivity and captures new phenomena relevant to the tails of the scalar distribution. These include approximately finite dispersion speeds arising at large Péclet number$\mathit{Pe}$(corresponding to small molecular diffusivity) and, for two-dimensional cellular flows, anisotropic dispersion. Explicit asymptotic results are obtained for shear flows in the limit of large$\mathit{Pe}$. (A companion paper, Part 2, is devoted to the large-$\mathit{Pe}$asymptotic treatment of cellular flows.) The predictions of large-deviation theory are compared with Monte Carlo simulations that estimate the tails of concentration accurately using importance sampling.

Contributors

Contributors

Large deviation eigenvalue density for the soft edge Laguerre and Jacobi β-ensembles

We analyze the eigenvalue density for the Laguerre and Jacobi β-ensembles in the cases that the corresponding exponents are extensive. In particular, we obtain the asymptotic expansion up to terms o(1), in the large deviation regime outside the limiting interval of support. As found in recent studies of the large deviation density for the Gaussian β-ensemble, and Laguerre β-ensemble with fixed exponent, there is a scaling from this asymptotic expansion to the right tail asymptotics for the distribution of the largest eigenvalue at the soft edge.

Many Flows Asymptotics for SMART Scheduling Policies

Scheduling policies that favor small jobs have received growing attention due to their superior performance with respect to mean delay, e.g., Shortest Remaining Processing Time (SRPT) and Preemptive Shortest Job First (PSJF). In this paper, we study the delay distribution of a generalization of the class of scheduling policies called SMART (because policies in it have “SMAll Response Times”), which includes SRPT, PSJF, and a range of practical variants, in a discrete-time queueing system under the many sources large deviations regime. Our analysis of SMART in this regime (large number of flows and large capacity) hinges on a novel two-dimensional (2-D) queueing framework that employs virtual queues and total ordering of jobs. We prove that all SMART policies have the same asymptotic delay distribution as SRPT, i.e., the delay distribution has the same decay rate. In addition, we illustrate the improvements SMART policies make over First Come First Serve (FCFS) and Processor Sharing (PS). Our 2-D queueing technique is generalizable to other policies as well. As an example, we show how the Foreground-Background (FB) policy can be analyzed using a 2-D queueing framework. FB is a policy, not contained in SMART, which manages to bias towards small jobs without knowing which jobs are small in advance.

On the Concentration of Large Deviations for Fat Tailed Distributions

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 is characterized by fat tailed increments, large fluctuations of 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.

Formula omitted] transience, and its applications to overload detection

When controlling communication networks, it is of crucial importance to have procedures that are capable of checking whether there are unanticipated load changes. In this paper we develop techniques for detecting such load changes, in a setting in which each connection consumes roughly the same amount of bandwidth (with VoIP as a leading example). For the situation of exponential holding times an explicit analysis can be performed in a large-deviations regime, leading to approximations of the test statistic of interest (and, in addition, to results for the transient of the M/ M/ ∞ queue, which are of independent interest). Since this procedure is applicable to exponential holding times only, and is rather numerically involved, we develop an approximate procedure for general holding times. In this procedure we record the number of trunks occupied at equidistant points in time Δ , 2 Δ , … , where Δ is chosen sufficiently large to safely assume that the samples are independent; this procedure is backed by results on the transient of the M / G / ∞ queue, thus complementing earlier results on relaxation times. The validity of the testing procedures is demonstrated through an extensive set of numerical experiments.

Estimating generalized Lyapunov exponents for products of random matrices

We discuss several techniques for the evaluation of the generalized Lyapunov exponents which characterize the growth of products of random matrices in the large-deviation regime. A Monte Carlo algorithm that performs importance sampling using a simple random resampling step is proposed as a general-purpose numerical method which is both efficient and easy to implement. Alternative techniques complementing this method are presented. These include the computation of the generalized Lyapunov exponents by solving numerically an eigenvalue problem, and some asymptotic results corresponding to high-order moments of the matrix products. Taken together, the techniques discussed in this paper provide a suite of methods which should prove useful for the evaluation of the generalized Lyapunov exponents in a broad range of applications. Their usefulness is demonstrated on particular products of random matrices arising in the study of scalar mixing by complex fluid flows.

On large deviation regimes for random media models

The focus of this article is on the different behavior of large deviations of random subadditive functionals above the mean versus large deviations below the mean in two random media models. We consider the point-to-point first passage percolation time $a_n$ on $\mathbb{Z}^d$ and a last passage percolation time $Z_n$. For these functionals, we have $\lim_{n\to\infty}\frac{a_n}{n}=\nu$ and $\lim_{n\to\infty}\frac{Z_n}{n}=\mu$. Typically, the large deviations for such functionals exhibits a strong asymmetry, large deviations above the limiting value are radically different from large deviations below this quantity. We develop robust techniques to quantify and explain the differences.

Central limit theorem and large deviations of the fading Wyner cellular model via product of random matrices theory

We apply the theory of products of random matrices to the analysis of multi-user communication channels similar to the Wyner model, which are characterized by short-range intra-cell broadcasting. We study fluctuations of the per-cell sum-rate capacity in the non-ergodic regime and provide results of the type of the central limit theorem (CLT) and large deviations (LD). Our results show that CLT fluctuations of the per-cell sum-rate C m are of order $$ 1/\sqrt m $$, where m is the number of cells, whereas they are of order 1/m in classical random matrix theory. We also show an LD regime of the form P(|C m − C| > ɛ) ≤ e −mα with α = α(ɛ) > 0 and C = $$ \mathop {\lim }\limits_{m \to \infty } $$ C m , as opposed to the rate $$ e^{ - m^2 \alpha } $$ in classical random matrix theory.

Large deviations in randomly coloured random graphs

Models of random graphs are considered where the presence or absence of an edge depends on the random types (colours) of its vertices, so that whether or not edges are present can be dependent. The principal objective is to study large deviations in the number of edges. These graphs provide a natural example with two different non-degenerate large deviation regimes, one arising from large deviations in the colourings followed by typical edge placement and the other from large deviation in edge placement. A secondary objective is to illustrate the use of a general result on large deviations for mixtures.

State-dependent importance sampling for regularly varying random walks

Consider a sequence (Xk:k≥ 0) of regularly varying independent and identically distributed random variables with mean 0 and finite variance. We develop efficient rare-event simulation methodology associated with large deviation probabilities for the random walk (Sn:n≥ 0). Our techniques are illustrated by examples, including large deviations for the empirical mean and path-dependent events. In particular, we describe two efficient state-dependent importance sampling algorithms for estimating the tail ofSnin a large deviation regime asn↗ ∞. The first algorithm takes advantage of large deviation approximations that are used to mimic the zero-variance change of measure. The second algorithm uses a parametric family of changes of measure based on mixtures. Lyapunov-type inequalities are used to appropriately select the mixture parameters in order to guarantee bounded relative error (or efficiency) of the estimator. The second example involves a path-dependent event related to a so-called knock-in financial option under heavy-tailed log returns. Again, the importance sampling algorithm is based on a parametric family of mixtures which is selected using Lyapunov bounds. In addition to the theoretical analysis of the algorithms, numerical experiments are provided in order to test their empirical performance.

State-dependent importance sampling for regularly varying random walks

Consider a sequence (X k : k ≥ 0) of regularly varying independent and identically distributed random variables with mean 0 and finite variance. We develop efficient rare-event simulation methodology associated with large deviation probabilities for the random walk (S n : n ≥ 0). Our techniques are illustrated by examples, including large deviations for the empirical mean and path-dependent events. In particular, we describe two efficient state-dependent importance sampling algorithms for estimating the tail of S n in a large deviation regime as n ↗ ∞. The first algorithm takes advantage of large deviation approximations that are used to mimic the zero-variance change of measure. The second algorithm uses a parametric family of changes of measure based on mixtures. Lyapunov-type inequalities are used to appropriately select the mixture parameters in order to guarantee bounded relative error (or efficiency) of the estimator. The second example involves a path-dependent event related to a so-called knock-in financial option under heavy-tailed log returns. Again, the importance sampling algorithm is based on a parametric family of mixtures which is selected using Lyapunov bounds. In addition to the theoretical analysis of the algorithms, numerical experiments are provided in order to test their empirical performance.

Large Deviations of Queues Sharing a Randomly Time-Varying Server

We consider a discrete-time model where multiple queues, each with its own exogenous arrival process, are served by a server whose capacity varies randomly and asynchronously with respect to different queues. This model is primarily motivated by the problem of efficient scheduling of transmissions of multiple data flows sharing a wireless channel. We address the following problem of controlling large deviations of the queues: find a scheduling rule, which is optimal in the sense of maximizing 0.1 $$\min_{i}\biggl[\lim_{n\to\infty}\frac{-1}{n}\log P(a_{i}Q_{i}>n)\biggr],$$ where Q i is the length of the i-th queue in a stationary regime, and a i >0 are parameters. Thus, we seek to maximize the minimum of the exponential decay rates of the tails of distributions of weighted queue lengths a i Q i . We give a characterization of the upper bound on (0.1) under any scheduling rule, and of the lower bound on (0.1) under the exponential (EXP) rule. We prove that the two bounds match, thus proving optimality of the EXP rule. The EXP rule is very parsimonious in that it does not require any "pre-computation" of its parameters, and uses only current state of the queues and of the server. The EXP rule is not invariant with respect to scaling of the queues, which complicates its analysis in the large deviations regime. To overcome this, we introduce and prove a refined sample path large deviations principle, or refined Mogulskii theorem, which is of independent interest.

Kraichnan Flow in a Square: An Example of Integrable Chaos

The Kraichnan flow provides an example of a random dynamical system accessible to an exact analysis. We study the evolution of the infinitesimal separation between two Lagrangian trajectories of the flow. Its long-time asymptotics is reflected in the multiplicative large deviation regime of the statistics of stretching exponents. In the flow in a two-dimensional periodic square where the anisotropy persists at small scales, the calculation of the large deviation rate function of the stretching exponents reduces to the study of the ground state energy of an integrable periodic Schroedinger operator of the Lame type. The underlying integrability permits to explicitly exhibit the non-Gaussianity of the multiplicative large deviations, to analyze the time-scales at which the large deviation regime sets in and to identify the general scenario for the appearance of multiplicative large deviations and the restrictions on its applicability.

Overcrowding estimates for zeroes of Planar and Hyperbolic Gaussian analytic functions

We consider the point process of zeroes of certain Gaussian analytic functions and find the asymptotics for the probability that there are more than m points of the process in a fixed disk of radius r, as $$r\rightarrow \infty$$ . For the planar Gaussian analytic function, $$\sum_{n \geq 0}\frac{a_n z^n}{\sqrt{n!}}$$ , we show that this probability is asymptotic to $${e^{-\frac{1}{2}m^2\log(m)}}$$ . For the hyperbolic Gaussian analytic functions, $${\sum_{n\geq 0}{-\rho \choose n}^{1/2} a_n z^n}, \rho > 0$$ , we show that this probability decays like $$e^{-cm^2}$$ . In the planar case, we also consider the problem posed by Mikhail Sodin2 on moderate and very large deviations in a disk of radius r, as $$r\rightarrow \infty$$ . We partially solve the problem by showing that there is a qualitative change in the asymptotics of the probability as we move from the large deviation regime to the moderate.

Some exactly solvable models of urn process theory

We establish a fundamental isomorphism between discrete-time balanced urn processes and certain ordinary differential systems, which are nonlinear, autonomous, and of a simple monomial form. As a consequence, all balanced urn processes with balls of two colours are proved to be analytically solvable in finite terms. The corresponding generating functions are expressed in terms of certain Abelian integrals over curves of the Fermat type (which are also hypergeometric functions), together with their inverses. A consequence is the unification of the analyses of many classical models, including those related to the coupon collector's problem, particle transfer (the Ehrenfest model), Friedman's "adverse campaign'' and Pólya's contagion model, as well as the OK Corral model (a basic case of Lanchester's theory of conflicts). In each case, it is possible to quantify very precisely the probable composition of the urn at any discrete instant. We study here in detail "semi-sacrificial'' urns, for which the following are obtained: a Gaussian limiting distribution with speed of convergence estimates as well as a characterization of the large and extreme large deviation regimes. We also work out explicitly the case of $2$-dimensional triangular models, where local limit laws of the stable type are obtained. A few models of dimension three or greater, e.g., "autistic'' (generalized Pólya), cyclic chambers (generalized Ehrenfest), generalized coupon-collector, and triangular urns, are also shown to be exactly solvable.

Explicit Solution for a Network Control Problem in the Large Deviation Regime

We consider optimal control of a stochastic network, where service is controlled to prevent buffer overflow. We use a risk-sensitive escape time criterion, which in comparison to the ordinary escape time criteria heavily penalizes exits which occur on short time intervals. A limit as the buffer sizes tend to infinity is considered. In [2] we showed that, for a large class of networks, the limit of the normalized cost agrees with the value function of a differential game. In this game, one player controls the service discipline (who to serve and whether to serve), and the other player chooses arrival and service rates in the network. The game's value is characterized in [2] as the unique solution to a Hamilton–Jacobi–Bellman Partial Differential Equation (PDE). In the current paper we apply this general theory to the important case of a network of queues in tandem. Our main results are: (i) the construction of an explicit solution to the corresponding PDE, and (ii) drawing out the implications for optimal risk-sensitive and robust regulation of the network. In particular, the following general principle can be extracted. To avoid buffer overflow there is a natural competition between two tendencies. One may choose to serve a particular queue, since that will help prevent its own buffer from overflowing, or one may prefer to stop service, with the goal of preventing overflow of buffers further down the line. The solution to the PDE indicates the optimal choice between these two, specifying the parts of the state space where each queue must be served (so as not to lose optimality), and where it can idle. Referring to those queues which must be served as bottlenecks, one can use the solution to the PDE to explicitly calculate the bottleneck queues as a function of the system's state, in terms of a simple set of equations.

Moderate Deviations for I.I.D. Random Variables

We derive necessary and sufficient conditions for a sum of i.i.d. random variables – where , but – to satisfy a moderate deviations principle. Moreover we show that this equivalence is a typical moderate deviations phenomenon. It is not true in a large deviations regime.

Moderate Deviations for Longest Increasing Subsequences: The Lower Tail

We derive a moderate deviation principle for the lower tail probabilities of the length of a longest increasing subsequence in a random permutation. It refers to the regime between the lower tail large deviation regime and the central limit regime. The present article together with the upper tail moderate deviation principle in Ref. 12 yields a complete picture for the whole moderate deviation regime. Other than in Ref. 12, we can directly apply estimates by Baik, Deift, and Johansson, who obtained a (non-standard) Central Limit Theorem for the same quantity.