"We compute analytically, for large N, the probability P(N+,N) that a N×N Wishart random matrix has N+ eigenvalues exceeding a threshold Nζ, including its large deviation tails. This probability plays a benchmark role when performing the principal component analysis of a large empirical data set. We find that P(N+,N)≈exp⁡[-βN2ψζ(N+/N)], where β is the Dyson index of the ensemble and ψζ(κ) is a rate function that we compute explicitly in the full range 0≤κ≤1 and for any ζ. The rate function ψζ(κ) displays a quadratic behavior modulated by a logarithmic singularity close to its minimum κ⋆(ζ). This is shown to be a consequence of a phase transition in an associated Coulomb gas problem. The variance Δ(N) of the number of relevant components is also shown to grow universally (independent of ζ) as Δ(N)∼(βπ2)-1ln⁡N for large N."

"A large deviation function mathematically characterizes the statistical property of atypical events. Recently, in non-equilibrium statistical mechanics, large deviation functions have been used to describe universal laws such as the fluctuation theorem. Despite such significance, large deviation functions have not been easily obtained in laboratory experiments. Thus, in order to understand the physical significance of large deviation functions, it is necessary to consider their experimental measurability in greater detail. This aspect of large deviation is discussed with the presentation of a future problem."

"We investigate the sparse sample goodness-of-fit problem, where the number of samples $n$ is smaller than the size of the alphabet $m$. The goal of this work is to find an appropriate criterion to analyze statistical tests in this setting. A suitable model for analysis is the high-dimensional model in which both $n$ and $m$ tend to infinity, and $n=o(m)$. We propose a new performance criterion based on large deviation analysis, which generalizes the classical error exponent applicable for large sample problems (in which $m=O(n)$). This new criterion provides insights that are not available from asymptotic consistency or CLT analysis. The main results are:
(i) The best achievable probability of error $P_e$ decays as $-log(P_e)=(n^2/m)(1+o(1))J$ for some $J>0$.
(ii) A well-known coincidence-based test attains the optimal generalized error exponent.
(iii) The widely used Pearson's chi-square test has J=0.
(iv) The contributions (i)-(iii) are established under the assumption that the distribution under the null hypothesis is uniform. For the non-uniform case, a new test is proposed, with a non-zero generalized error exponent."

"On a finite inhomogeneous graph model for complex network, we define the Ising model, which is a paradigm model in statistical mechanics. For the ferromagnetic Ising model,we calculate the Thermodynamic limit of pressure per particle. From our results, we compute other physical quantities such as the magnetization and susceptibility, and investigate the critical behaviour of this model. Our calculations use large deviation principles (developed recently) for suitably defined empirical neighbourhood measures on inhomogeneous random graph."

"A large deviations principle is established for the joint law of the empirical measure and the flow measure of a renewal Markov process on a finite graph. We do not assume any bound on the arrival times, allowing heavy tailed distributions. In particular, the rate functional is in general degenerate (it has a nontrivial set of zeros) and not strictly convex. These features show a behavior highly different from what one may guess with a heuristic Donsker-Varadhan analysis of the problem."

"Stochastic partial differential equations driven by Poisson random measures (PRM) have been proposed as models for many different physical systems, where they are viewed as a refinement of a corresponding noiseless partial differential equations (PDE). A systematic framework for the study of probabilities of deviations of the stochastic PDE from the deterministic PDE is through the theory of large deviations. The goal of this work is to develop the large deviation theory for small Poisson noise perturbations of a general class of deterministic infinite dimensional models. Although the analogous questions for finite dimensional systems have been well studied, there are currently no general results in the infinite dimensional setting. This is in part due to the fact that in this setting solutions may have little spatial regularity, and thus classical approximation methods for large deviation analysis become intractable. The approach taken here, which is based on a variational representation for nonnegative functionals of general PRM, reduces the proof of the large deviation principle to establishing basic qualitative properties for controlled analogues of the underlying stochastic system. As an illustration of the general theory, we consider a particular system that models the spread of a pollutant in a waterway."

"Sequential Monte Carlo methods which involve sequential importance sampling and resampling are shown to provide a versatile approach to computing probabilities of rare events. By making use of martingale representations of the sequential Monte Carlo estimators, we show how resampling weights can be chosen to yield logarithmically efficient Monte Carlo estimates of large deviation probabilities for multidimensional Markov random walks."

"For fat tailed distributions (i.e. those that decay slower than an exponential), large deviations not only become relatively likely, but the way in which they are realized changes dramatically: A finite fraction of the whole sample deviation is concentrated on a single variable: large deviations are not the accumulation of many small deviations, but rather they are dominated to a single large fluctuation. 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 {em spontaneously broken}. This phenomenon has been discussed in the context of mass transport models in physics, where it takes the form of a condensation phase transition. Yet, the phenomenon is way more general. For example, in risk management of large portfolios, it suggests that one should expect losses to concentrate on a single asset: when extremely bad things happen, it is likely that there is a single factor on which bad luck concentrates. Along similar lines, one should expect that bubbles in financial markets do not gradually deflate, but rather burst abruptly and that in the most rainy day of a year, precipitation concentrate on a given spot. Analogously, when applied to biological evolution, we're lead to infer that, if fitness changes for individual mutations have a broad distribution, those large deviations that lead to better fit species are not likely to result from the accumulation of small positive mutations. Rather they are likely to arise from large rare jumps."

"We study large deviation properties of systems of weakly interacting particles modeled by Itô stochastic differential equations (SDEs). It is known under certain conditions that the corresponding sequence of empirical measures converges, as the number of particles tends to infinity, to the weak solution of an associated McKean–Vlasov equation. We derive a large deviation principle via the weak convergence approach. The proof, which avoids discretization arguments, is based on a representation theorem, weak convergence and ideas from stochastic optimal control. The method works under rather mild assumptions and also for models described by SDEs not of diffusion type. To illustrate this, we treat the case of SDEs with delay."

"The large-deviation method allows to characterize an ergodic counting process in terms of a thermodynamic frame where a free energy function determines the asymptotic non-stationary statistical properties of its fluctuations. Here, we study this formalism through a statistical mechanics approach, i.e., with an auxiliary counting process that maximizes an entropy function associated to the thermodynamic potential. We show that the realizations of this auxiliary process can be obtained after applying a conditional measurement scheme to the original ones, providing is this way an alternative measurement interpretation of the thermodynamic approach. General results are obtained for renewal counting processes, i.e., those where the time intervals between consecutive events are independent and defined by a unique waiting time distribution. The underlying statistical mechanics is controlled by the same waiting time distribution, rescaled by an exponential decay measured by the free energy function. A scale invariance, shift closure, and intermittence phenomena are obtained and interpreted in this context. Similar conclusions apply for non-renewal processes when the memory between successive events is induced by a stochastic waiting time distribution."

"I give a brief overview of the resolution of the apparent problem of reconciling time symmetric microscopic dynamic with time asymmetric equations describing the evolution of macroscopic variables. I then show how the large deviation function of the stationary state of the microscopic system can be used as a Lyapunov function for the macroscopic evolution equations."

"The growing availability of network data and of scientific interest in distributed systems has led to the rapid development of statistical models of network structure. Typically, however, these are models for the entire network, while the data consists only of a sampled sub-network. Parameters for the whole network, which is what is of interest, are estimated by applying the model to the sub-network. This assumes that the model is consistent under sampling, or, in terms of the theory of stochastic processes, that it defines a projective family. Focussing on the popular class of exponential random graph models (ERGMs), we show that this apparently trivial condition is in fact violated by many popular and scientifically appealing models, and that satisfying it drastically limits ERGM's expressive power. These results are actually special cases of more general ones about exponential families of dependent random variables, which we also prove. Using such results, we offer easily checked conditions for the consistency of maximum likelihood estimation in ERGMs, and discuss some possible constructive responses."

"Large and moderate deviation probabilities play an important role in many applied areas, such as insurance and risk analysis. This paper studies the exact moderate and large deviation asymptotics in non-logarithmic form for linear processes with independent innovations. The linear processes we analyze are general and therefore they include the long memory case. We give an asymptotic representation for probability of the tail of the normalized sums and specify the zones in which it can be approximated either by a standard normal distribution or by the marginal distribution of the innovation process. The results are then applied to regression estimates, moving averages, fractionally integrated processes, linear processes with regularly varying exponents and functions of linear processes. We also consider the computation of value at risk and expected shortfall, fundamental quantities in risk theory and finance."

"Classical Edgeworth expansions provide asymptotic correction terms to the Central Limit Theorem (CLT) up to an order that depends on the number of moments available. In this paper, we provide subsequent correction terms beyond those given by a standard Edgeworth expansion in the general case of regularly varying distributions with diverging moments (beyond the second). The subsequent terms can be expressed in a simple closed form in terms of certain special functions (Dawson’s integral and parabolic cylinder functions), and there are qualitative differences depending on whether the number of moments available is even, odd, or not an integer, and whether the distributions are symmetric or not. If the increments have an even number of moments, then additional logarithmic corrections must also be incorporated in the expansion parameter. An interesting feature of our correction terms for the CLT is that they become dominant outside the central region and blend naturally with known large-deviation asymptotics when these are applied formally to the spatial scales of the CLT."

"The theory of large deviations has been applied successfully in the last 30 years or so to study the properties of equilibrium systems and to put the foundations of equilibrium statistical mechanics on a clearer and more rigorous footing. A similar approach has been followed more recently for nonequilibrium systems, especially in the context of interacting particle systems. We review here the basis of this approach, emphasizing the similarities and differences that exist between the application of large deviation theory for studying equilibrium systems on the one hand and nonequilibrium systems on the other. Of particular importance are the notions of macroscopic, hydrodynamic, and long-time limits, which are analogues of the equilibrium thermodynamic limit, and the notion of statistical ensembles which can be generalized to nonequilibrium systems. For the purpose of illustrating our discussion, we focus on applications to Markov processes, in particular to simple random walks."

"We derive the differential equation describing the time evolution of the work probability distribution function of a stochastic system which is driven out of equilibrium by the manipulation of a parameter. We consider both systems described by their microscopic state or by a collective variable which identifies a quasiequilibrium state. We show that the work probability distribution can be represented by a path integral, which is dominated by “classical” paths in the large system size limit. We compare these results with simulated manipulation of mean-field systems. We discuss the range of applicability of the Jarzynski equality for evaluating the system free energy using these out-of-equilibrium manipulations. Large fluctuations in the work and the shape of the work distribution tails are also discussed."

"We discuss entropy production in nonequilibrium steady states by focusing on paths obtained by sampling at regular (small) intervals, instead of sampling on each change of the system’s state. This allows us to directly study entropy production in systems with microscopic irreversibility. The two sampling methods are equivalent otherwise, and the fluctuation theorem also holds for the different paths. We focus on a fully irreversible three-state loop, as a canonical model of microscopic irreversibility, finding its entropy distribution, rate of entropy production, and large deviation function in closed analytical form, and showing that the observed kink in the large deviation function arises solely from microscopic irreversibility."

"In this paper we derive the moderate deviation principle for stationary sequences of bounded random variables under martingale-type conditions. Applications to functions of $phi$-mixing sequences, contracting Markov chains, expanding maps of the interval, and symmetric random walks on the circle are given."

Useful 3 pp. cheat-sheet

In a public form, at last.

Very cool.

I'd mark it "to read", but it's in French!

Rigorous results on conditions under which mean field approximations become exact for large ERGMs, and consequently they end up looking like Erdos-Renyi graphs (at some effective density); these come from clever large-deviations arguments.  To over-simplify some pretty theorems, the cases are (1) all interactions between edges are positive, so introducing any positive density of edges at all tends to produce a blow-up towards a nearly complete graph (where edges are independent); or (2) all interactions are extremely weak, and consequently negligible in the large-scale limit.

"In this paper we prove that the stationary distribution of populations in genetic algorithms focuses on the uniform population with the highest fitness value as the selective pressure goes to ∞ and the mutation probability goes to 0. The obtained sufficient condition is based on the work of Albuquerque and Mazza (2000), who, following Cerf (1998), applied the large deviation principle approach (Freidlin-Wentzell theory) to the Markov chain of genetic algorithms. The sufficient condition is more general than that of Albuquerque and Mazza, and covers a set of parameters which were not found by Cerf."

"We discuss research done in two important areas of nonequilibrium statistical mechanics: fluctuation dissipation relations and dynamical fluctuations. In equilibrium systems the fluctuation-dissipation theorem gives a simple relation between the response of observables to a perturation and correlation functions in the unperturbed system. Our contribution here is an investigation of the form of the response function for systems out of equilibrium. Furthermore, we use the theory of large deviations to examine dynamical fluctuations in systems out of equilibrium. In dynamical fluctuation theory we consider two kinds of observables: occupations (describing the fraction of time the system spends in each configuration) and currents (describing the changes of configuration the system makes). We explain how to compute the rate functions of the large deviations, and what the physical quantities are that govern their form."

"This paper proposes a new notion of typical sequences on a wide class of abstract alphabets (so-called standard Borel spaces), which is based on approximations of memoryless sources by empirical distributions uniformly over a class of measurable "test functions." In the finite-alphabet case, we can take all uniformly bounded functions and recover the usual notion of strong typicality (or typicality under the total variation distance). For a general alphabet, however, this function class turns out to be too large, and must be restricted. With this in mind, we define typicality with respect to any Glivenko-Cantelli function class (i.e., a function class that admits a Uniform Law of Large Numbers)..."

"how to extract a hypothesis with small risk from the ensemble of hypotheses generated by an arbitrary on-line learning algorithm run on [IID data]. ... a simple large deviation argument [proves] tight data-dependent bounds for the risk of this hypothesis in terms of an easily computable statistic Mn associated with the on-line performance of the ensemble. Via sharp pointwise bounds on Mn, we then obtain risk tail bounds for kernel perceptron algorithms in terms of the spectrum of the empirical kernel matrix. ... A distinctive feature of our approach is that the key tools for our analysis come from the model of prediction of individual sequences; i.e., a model making no probabilistic assumptions on the source generating the data. In fact, these tools turn out to be so powerful that we only need very elementary statistical facts to obtain our final risk bounds." Bounced off this 2004; try again.

"The paper focuses on general properties of parametric minimum contrast estimators. The quality of estimation is measured in terms of the rate function related to the contrast, thus allowing to derive exponential risk bounds invariant with respect to the detailed probabilistic structure of the model. This approach works well for small or moderate samples and covers the case of a misspecified parametric model. Another important feature of the presented bounds is that they may be used in the case when the parametric set is unbounded and non-compact. These bounds do not rely on the entropy or covering numbers and can be easily computed. The most important statistical fact resulting from the exponential bonds is a concentration inequality which claims that minimum contrast estimators concentrate with a large probability on the level set of the rate function. In typical situations, every such set is a root-n neighborhood of the parameter of interest."

Very clever, though they do not derive explicit rates. --- I wonder if this couldn't be shown as a consequence of general properties of very-weak-Bernoulli processes? The vwB property would seem to imply perfect simulation, but I am not sure about the converse.

"This paper reports on a macroscopic fluctuation theory developed over the last ten years in collaboration with L. Bertini, A. De Sole, D. Gabrielli and C. Landim. This theory has been inspired by and tested on stochastic models of interacting particles (stochastic lattice gases). It is the basis for a new approach to the study of stationary non equilibrium states applicable to a large class of systems. This overview emphasizes general ideas and for the details I refer to the published papers."

"This is a survey of recent developments in the area of transport inequalities. We investigate their consequences in terms of concentration and deviation inequalities and sketch their links with other functional inequalities and also large deviation theory."

"the extent to which truncated heavy tailed random vectors retain the characteristic features of heavy tailed random vectors, is answered from the point of views of the central limit theorem and the large deviations behavior. The analysis of the central limit behavior of the partial sums of observations coming from a heavy-tailed model is done for random vectors taking values in a separable Banach space. For the large deviations analysis, the random vectors are assumed to be R^d-valued. It turns out that there are two regimes depending on the growth rate of the truncating threshold, so that in one regime, much of the heavy tailedness is retained, while in the other regime, the same is lost."

"the rate function is exactly that of the averaged equation plus the fluctuating deviation which is a stochastic partial differential equation with small Gaussian perturbation"

With applications promised to SDEs.

"We consider a partially hyperbolic set $K$ on a Riemannian manifold $M$ whose tangent space splits as $T_K M=E^{cu}\oplus E^{s}$, for which the centre-unstable direction $E^{cu}$ expands non-uniformly on some local unstable disk. We show that under these assumptions $f$ induces a Gibbs-Markov structure. Moreover, the decay of the return time function can be controlled in terms of the time typical points need to achieve some uniform expanding behavior in the centre-unstable direction. As an application of the main result we obtain certain rates for decay of correlations, large deviations, an almost sure invariance principle and the validity of the Central Limit Theorem."

"Onsager-Machlup-type theory for nonequilibrium semi-Markov processes. Our main result is an exact large time asymptotics for the joint probability of the occupation times and the currents in the system, establishing some generic large deviation structures. We discuss in detail how the nonequilibrium driving and the non-exponential waiting time distribution influence the occupation-current statistics. The violation of the Markov condition is reflected in the emergence of a new type of nonlocality in the fluctuations. Explicit solutions are obtained for some examples of driven random walks on the ring."

"This paper concerns the statistical properties of hyperbolic diffeomorphisms. We obtain a large deviation result with respect to slowly shrinking intervals for a large class of Hölder continuous functions. In case of time reversal symmetry, we obtain a corresponding version of the Fluctuation Theorem."

Mine for examples for _Almost None_?

Looks like a teaser for the book by Brazzale, Davison and Reid.

"We claim that looking at probability distributions of emph{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 Poincar'e recurrences in the -quite delicate- case of dynamical systems with weak chaotic properties."

I'm pretty impressed.

Tutorial review paper. Really excellent; made me want to do stat. mech. again. (I will, as my grandfather used to say, lie down until the temptation passes.)

not-quite-final PDF draft of the classic book

One of the creators surveys the field. Wald Lecture for 2005.

Suitable as an application for Almost None, maybe?