Large Deviation Properties Research Articles

Thermodynamic large fluctuations from uniformized dynamics

Large fluctuations have received considerable attention as they encode information on the fine-scale dynamics. Large deviation relations known as fluctuation theorems also capture crucial nonequilibrium thermodynamical properties. Here we report that, using the technique of uniformization, the thermodynamic large deviation functions of continuous-time Markov processes can be obtained from Markov chains evolving in discrete time. This formulation offers theoretical and numerical approaches to explore large deviation properties. In particular, the time evolution of autonomous and nonautonomous processes can be expressed in terms of a single Poisson rate. In this way the uniformization procedure leads to a simple and efficient way to simulate stochastic trajectories that reproduce the exact fluxes statistics. We illustrate the formalism for the current fluctuations in a stochastic pump model.

Large deviation principle for semilinear stochastic evolution equations with monotone nonlinearity and multiplicative noise

We demonstrate the large deviation property for the mild solutions of stochastic evolution equations with monotone nonlinearity and multiplicative noise. This is achieved using the recently developed weak convergence method, in studying the large deviation principle. An Itô-type inequality is a main tool in the proofs. We also give two examples to illustrate the applications of the theorems.

Non-Equilibrium Thermodynamics and Topology of Currents

In many experimental situations, a physical system undergoes stochastic evolution which may be described via random maps between two compact spaces. In the current work, we study the applicability of large deviations theory to time-averaged quantities which describe such stochastic maps, in particular time-averaged currents and density functionals. We derive the large deviations principle for these quantities, as well as for global topological currents, and formulate variational, thermodynamic relations to establish large deviation properties of the topological currents. We illustrate the theory with a nontrivial example of a Heisenberg spin-chain with a topological driving of the Wess-Zumino type. The Cram\'er functional of the topological current is found explicitly in the instanton gas regime for the spin-chain model in the weak-noise limit. In the context of the Morse theory, we discuss a general reduction of continuous stochastic models with weak noise to effective Markov chains describing transitions between stable fixed points.

Instability statistics and mixing rates

We claim that looking at probability distributions of finite time largest Lyapunov exponents, and more precisely studying their large deviation properties, yields an extremely powerful technique to get quantitative estimates of polynomial decay rates of time correlations and Poincaré recurrences in the-quite-delicate case of dynamical systems with weak chaotic properties.

Canonical Moments and Random Spectral Measures

We study some connections between the random moment problem and random matrix theory. A uniform draw in a space of moments can be lifted into the spectral probability measure of the pair (A,e), where A is a random matrix from a classical ensemble, and e is a fixed unit vector. This random measure is a weighted sampling among the eigenvalues of A. We also study the large deviations properties of this random measure when the dimension of the matrix increases. The rate function for these large deviations involves the reversed Kullback information.

Martin boundary of a reflected random walk on a half-space

The complete representation of the Martin compactification for reflected random walks on a half-space $${\mathbb{Z}^d\times\mathbb{N}}$$ is obtained. It is shown that the full Martin compactification is in general not homeomorphic to the “radial” compactification obtained by Ney and Spitzer for the homogeneous random walks in $${\mathbb{Z}^d}$$ : convergence of a sequence of points $${z_n\in\mathbb{Z}^{d-1}\times\mathbb{N}}$$ to a point of on the Martin boundary does not imply convergence of the sequence z n /|z n | on the unit sphere S d . Our approach relies on the large deviation properties of the scaled processes and uses Pascal’s method combined with the ratio limit theorem. The existence of non-radial limits is related to non-linear optimal large deviation trajectories.

Large deviation properties for empirical quantile-type production functions

In standard microeconomic theory of the firm, the production frontier is used to describe the geometric locus of the optimal production. Most limitations of the conventional estimators of this efficient frontier stem from its reliance on estimation of the upper surface of the joint support of (X, Y), where X∈ℝ p + represents the inputs-usage and Y∈ℝ+ is the produced output. Instead, a recent approach involving the estimation of a partial quantile-type frontier of the joint support, lying close to its full boundary, has emerged in the literature as an attractive alternative. In this paper, we present a new and simple proof of the asymptotic normality of the resulting nonparametric frontier estimators and provide an estimate for the accuracy of the normal approximation. We also propose a Kiefer-type asymptotic representation for these estimators. The main results concern the study of some weak, moderate and strong large deviation properties of the quantile-based frontier estimators.

Sharp Large Deviations for the Fractional Ornstein–Uhlenbeck Process

We investigate the sharp large deviation properties of the energy and the maximum likelihood estimator for the Ornstein-Uhlenbeck process driven by a fractional Brownian motion with Hurst index greater than one half.

Percolation transition and distribution of connected components in generalized random network ensembles

In this work, we study the percolation transition and large deviation properties of generalized canonical network ensembles. This new type of random networks might have a very rich complex structure, including high heterogeneous degree sequences, non-trivial community structure or specific spatial dependence of the link probability for networks embedded in a metric space. We find the cluster distribution of the networks in these ensembles by mapping the problem to a fully connected Potts model with heterogeneous couplings. We show that the nature of the Potts model phase transition, linked to the birth of a giant component, has a crossover from second to first order when the number of critical colors qc = 2 in all the networks under study. These results shed light on the properties of dynamical processes defined on these network ensembles.

Weighted Poincaré-type inequalities for Cauchy and other convex measures

Brascamp–Lieb-type, weighted Poincaré-type and related analytic inequalities are studied for multidimensional Cauchy distributions and more general κ-concave probability measures (in the hierarchy of convex measures). In analogy with the limiting (infinite-dimensional log-concave) Gaussian model, the weighted inequalities fully describe the measure concentration and large deviation properties of this family of measures. Cheeger-type isoperimetric inequalities are investigated similarly, giving rise to a common weight in the class of concave probability measures under consideration.

Large deviations for eigenvalues of sample covariance matrices, with applications to mobile communication systems

We study sample covariance matrices of the form W = (1 / n)CCT, where C is a k x n matrix with independent and identically distributed (i.i.d.) mean 0 entries. This is a generalization of the so-called Wishart matrices, where the entries of C are i.i.d. standard normal random variables. Such matrices arise in statistics as sample covariance matrices, and the high-dimensional case, when k is large, arises in the analysis of DNA experiments. We investigate the large deviation properties of the largest and smallest eigenvalues of W when either k is fixed and n → ∞ or kn → ∞ with kn = o(n / log log n), in the case where the squares of the i.i.d. entries have finite exponential moments. Previous results, proving almost sure limits of the eigenvalues, require only finite fourth moments. Our most explicit results for large k are for the case where the entries of C are ∓ 1 with equal probability. We relate the large deviation rate functions of the smallest and largest eigenvalues to the rate functions for i.i.d. standard normal entries of C. This case is of particular interest since it is related to the problem of decoding of a signal in a code-division multiple-access (CDMA) system arising in mobile communication systems. In this example, k is the number of users in the system and n is the length of the coding sequence of each of the users. Each user transmits at the same time and uses the same frequency; the codes are used to distinguish the signals of the separate users. The results imply large deviation bounds for the probability of a bit error due to the interference of the various users.

Large deviations for eigenvalues of sample covariance matrices, with applications to mobile communication systems

We study sample covariance matrices of the form W = (1 / n) C C T, where C is a k x n matrix with independent and identically distributed (i.i.d.) mean 0 entries. This is a generalization of the so-called Wishart matrices, where the entries of C are i.i.d. standard normal random variables. Such matrices arise in statistics as sample covariance matrices, and the high-dimensional case, when k is large, arises in the analysis of DNA experiments. We investigate the large deviation properties of the largest and smallest eigenvalues of W when either k is fixed and n → ∞ or k n → ∞ with k n = o(n / log log n), in the case where the squares of the i.i.d. entries have finite exponential moments. Previous results, proving almost sure limits of the eigenvalues, require only finite fourth moments. Our most explicit results for large k are for the case where the entries of C are ∓ 1 with equal probability. We relate the large deviation rate functions of the smallest and largest eigenvalues to the rate functions for i.i.d. standard normal entries of C . This case is of particular interest since it is related to the problem of decoding of a signal in a code-division multiple-access (CDMA) system arising in mobile communication systems. In this example, k is the number of users in the system and n is the length of the coding sequence of each of the users. Each user transmits at the same time and uses the same frequency; the codes are used to distinguish the signals of the separate users. The results imply large deviation bounds for the probability of a bit error due to the interference of the various users.

Publisher's Note: “Large Deviation Property of Free Energy in p-Body Sherrington–Kirkpatrick Model”

Publisher's Note: “Large Deviation Property of Free Energy in <i>p</i>-Body Sherrington–Kirkpatrick Model”

Large Deviation Property of Free Energy inp-Body Sherrington–Kirkpatrick Model

Cumulant generating function phi(n) and rate function Sigma(f) of the free energy is evaluated in p-body Sherrington-Kirkpatrick model by using the replica method with the replica number n finite. From a perturbational argument, we show that the cumulant generating function is constant in the vicinity of n = 0. On the other hand, with the help of two analytic properties of phi(n), the behavior of phi(n) is derived again. However this is also shown to be broken at a finite value of n, which gives a characteristic value in the rate function near the thermodynamic value of the free energy. Through the continuation of phi(n) as a function of n, we find out a way to derive the 1RSB solution at least in this model, which is to fix the RS solution to be a monotone increasing function.

Large deviations for infinite dimensional stochastic dynamical systems

The large deviations analysis of solutions to stochastic differential equations and related processes is often based on approximation. The construction and justification of the approximations can be onerous, especially in the case where the process state is infinite dimensional. In this paper we show how such approximations can be avoided for a variety of infinite dimensional models driven by some form of Brownian noise. The approach is based on a variational representation for functionals of Brownian motion. Proofs of large deviations properties are reduced to demonstrating basic qualitative properties (existence, uniqueness and tightness) of certain perturbations of the original process.

NETWORK AS A CHAOTIC DYNAMICAL SYSTEM

A directed network such as the WWW can be represented by a transition matrix. Comparing this matrix to a Frobenius–Perron matrix of a chaotic piecewise-linear one-dimensional map whose domain can be divided into Markov subintervals, we are able to relate the network structure itself to chaotic dynamics. Just like various large deviation properties of local expansion rates (finite-time Lyapunov exponents) related to chaotic dynamics, we can also discuss those properties of network structure.

Large deviation properties of constant rate data streams sharing a buffer with long-range dependent traffic in critical loading

We consider a constant rate traffic which shares a buffer with a random cross traffic. A first come first served or priority service discipline is applied at the buffer. After service at the first buffer the constant rate traffic moves to a play-out buffer. Both buffers provide service at constant rate and infinite waiting room. We investigate logarithmic large and moderate deviation asymptotics for the tail probabilities of the steady-state queue length distribution at the play-out buffer for long-range dependent cross traffic in critical loading. We characterize the asymptotic behavior of the cross traffic which leads to a large queue length at the play-out buffer and compare it to the one for renewal cross traffic.

Large deviation properties of constant rate data streams sharing a buffer with long-range dependent traffic in critical loading

We consider a constant rate traffic which shares a buffer with a random cross traffic. A first come first served or priority service discipline is applied at the buffer. After service at the first buffer the constant rate traffic moves to a play-out buffer. Both buffers provide service at constant rate and infinite waiting room. We investigate logarithmic large and moderate deviation asymptotics for the tail probabilities of the steady-state queue length distribution at the play-out buffer for long-range dependent cross traffic in critical loading. We characterize the asymptotic behavior of the cross traffic which leads to a large queue length at the play-out buffer and compare it to the one for renewal cross traffic.

Temporal ordering of nonequilibrium fluctuations as a corollary of the second law of thermodynamics

In nonequilibrium steady states, the probabilities of the time-reversed paths differ in general from the path probabilities and they decrease at different rates given, respectively, by the time-reversed and standard entropies per unit time. These quantities characterize temporal disorder in the time-reversed paths and the paths themselves. The difference between these quantities gives the thermodynamic entropy production in units of Boltzmann's constant. Out of equilibrium, the temporal disorder of the time-reversed paths turns out to be larger than the temporal disorder of the paths themselves. In this way, a theorem is obtained according to which the temporal order of the paths is greater than for the time-reversed paths. The temporal disorder or dynamical randomness of the paths and their time-reversal have large-deviation properties directly connected to the thermodynamic entropy production. To cite this article: P. Gaspard, C. R. Physique 8 (2007).

Annealed deviations of random walk in random scenery

Let ( Z n ) n ∈ N be a d-dimensional random walk in random scenery, i.e., Z n = ∑ k = 0 n − 1 Y ( S k ) with ( S k ) k ∈ N 0 a random walk in Z d and ( Y ( z ) ) z ∈ Z d an i.i.d. scenery, independent of the walk. The walker's steps have mean zero and some finite exponential moments. We identify the speed and the rate of the logarithmic decay of P ( 1 n Z n > b n ) for various choices of sequences ( b n ) n in [ 1 , ∞ ) . Depending on ( b n ) n and the upper tails of the scenery, we identify different regimes for the speed of decay and different variational formulas for the rate functions. In contrast to recent work [A. Asselah, F. Castell, Large deviations for Brownian motion in a random scenery, Probab. Theory Related Fields 126 (2003) 497–527] by A. Asselah and F. Castell, we consider sceneries unbounded to infinity. It turns out that there are interesting connections to large deviation properties of self-intersections of the walk, which have been studied recently by X. Chen [X. Chen, Exponential asymptotics and law of the iterated logarithm for intersection local times of random walks, Ann. Probab. 32 (4) 2004].