"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."

"This paper develops goodness of fit statistics that can be used to formally assess Markov random field models for spatial data, when the model distributions are discrete or continuous and potentially parametric. Test statistics are formed from generalized spatial residuals which are collected over groups of nonneighboring spatial observations, called concliques. Under a hypothesized Markov model structure, spatial residuals within each conclique are shown to be independent and identically distributed as uniform variables. The information from a series of concliques can be then pooled into goodness of fit statistics. Under some conditions, large sample distributions of these statistics are explicitly derived for testing both simple and composite hypotheses, where the latter involves additional parametric estimation steps. The distributional results are verified through simulation, and a data example illustrates the method for model assessment."

"We study an extension to general Markov random fields of the resampling scheme given in Bickel and Levina (2006) [4] for texture synthesis with stationary Markov mesh models. The procedure generates bootstrap replicates of a sample using kernel regression and the principle of Gibbs sampling. Consistency of the bootstrap distribution is investigated under the Dobrushin contraction condition. Some simulation examples are given, in particular for the texture synthesis, for which the multiscale algorithm of Paget and Longstaff (1998) [27] is revisited."

"The present paper has two goals. First to present a natural example of a new class of random fields which are the variable neighborhood random fields. The example we consider is a partially observed nearest neighbor binary Markov random field. The second goal is to establish sufficient conditions ensuring that the variable neighborhoods are almost surely finite. We discuss the relationship between the almost sure finiteness of the interaction neighborhoods and the presence/absence of phase transition of the underlying Markov random field. In the case where the underlying random field has no phase transition we show that the finiteness of neighborhoods depends on a specific relation between the noise level and the Dobrushin coefficient. The case in which there is phase transition is addressed in the frame of the ferromagnetic Ising model. We prove that the existence of infinite interaction neighborhoods depends on the phase. The first result has a probabilistic proof using a Kalikow type decomposition of a Glauber dynamics associated to the field. The second result is proved using cluster expansion."

An attempt on the "n=1" problem. Alternative home: http://www.cs.purdue.edu/homes/neville/papers/xiang-neville-aistat2011.pdf

"This paper deals with the problem of perfect sampling from a Gibbs measure with infinite range interactions. We present some sufficient conditions for the extinction of processes which are like supermartingales when large values are taken. This result has profound consequences on perfect simulation, showing that local modifications on the interactions of a model do not affect the simulability. We also pose the question to optimize over a sequence of sets and we completely solve the question in the case of finite range interactions."

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

"We establish a central limit theorem and an invariance principle for stationary random fields. In particular, we extend the Maxwell--Woodroofe condition on stationary processes to the multiparameter setting. Our result is obtained via an $m$-dependent approximation method. As applications, we improve known results on the invariance principles for orthomartingales and functionals of linear random fields."

"Disordered systems are an important class of models in statistical mechanics, having the defining characteristic that the energy landscape is a fixed realization of a random field. Examples include various models of glasses and polymers. They also arise in other areas, like fitness models in evolutionary biology. The ground state of a disordered system is the state with minimum energy. The system is said to be chaotic if a small perturbation of the energy landscape causes a drastic shift of the ground state. We present a rigorous theory of chaos in disordered systems that confirms long-standing physics intuition about connections between chaos, anomalous fluctuations of the ground state energy, and the existence of multiple valleys in the energy landscape."

"Who then is my neighbor?" (Not an actual quote from the paper.)

I am not sure whether there shouldn't be a chapter on diagrammatic methods in _Almost None_, because I don't really understand how what I learned about them as a physicist relates to actual probability theory.  I will come back to this one later.

Found the pre-print, which I'd read in '04, while looking for something else in my office... Note that this is the same shape of mesh that Lindgren and Nordahl advocated for use in 2D information theory, on totally different (I think) grounds.

"Various methods of summation for divergent series of real numbers have been generalized to analogous results for sums of i.i.d. random variables. The natural extension of results corresponding to Cesàro summation amounts to proving almost sure convergence of the Cesàro means. In the present paper we extend such results as well as weak laws and results on complete convergence to random fields, more specifically to random variables indexed by ℤ+2, the positive two-dimensional integer lattice points."

"A Central Limit Theorem is proved for linear random fields when sums are taken over finite disjoint union of rectangles. The approach does not rely upon the use of Beveridge Nelson decomposition and the conditions needed are similar to those given by Ibragimov for linear processes. When specializing this result to the case when sums are being taken over rectangles, an analogue of Ibragimov result is obtained with a lot of uniformity."

"his introduction to some of the principal models in the theory of disordered systems leads the reader through the basics, to the very edge of contemporary research, with the minimum of technical fuss. Topics covered include random walk, percolation, self-avoiding walk, interacting particle systems, uniform spanning tree, random graphs, as well as the Ising, Potts, and random-cluster models for ferromagnetism, and the Lorentz model for motion in a random medium. Schramm–Löwner evolutions (SLE) arise in various contexts. The choice of topics is strongly motivated by modern applications and focuses on areas that merit further research. Special features include a simple account of Smirnov's proof of Cardy's formula for critical percolation, and a fairly full account of the theory of influence and sharp-thresholds. Accessible to a wide audience of mathematicians and physicists, this book can be used as a graduate course text. Each chapter ends with a range of exercises."

"The joint distribution of a d-dimensional random field restricted to a box of size k can be estimated by looking at a realization in a box of size $n \gg k$ and computing the empirical distribution. This is done by sliding a box of size k around in the box of size n and computing frequencies. We show that when $k = k(n)$ grows as a function of n, then the total variation distance between this empirical distribution and the true distribution goes to 0 a.s. as $n \to \infty$ provided $k(n)^d \leq (\log n^d)/(H + \varepsilon)$ (where H is the entropy of the random field) and providing the random field satisfies a condition called quite weak Bernoulli with exponential rate. ... Marton and Shields have proved such results in one dimension and this paper is an attempt to extend their results to some extent to higher dimensions."

"We consider the problem of estimating the topology of spatial interactions in a discrete state, discrete time spatio-temporal graphical model where the interactions affect the temporal evolution of each agent in a network. Among other models, the susceptible, infected, recovered ($SIR$) model for interaction events fall into this framework. We pose the problem as a structure learning problem and solve it using an $\ell_1$-penalized likelihood convex program. We evaluate the solution on a simulated spread of infectious over a complex network. Our topology estimates outperform those of a standard spatial Markov random field graphical model selection using $\ell_1$-regularized logistic regression."

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.

"We study the nonparametric covariance estimation of a stationary Gaussian field X observed on a regular lattice. In the time series setting, some procedures ... achieve optimal model selection among autoregressive models. ... no such equivalent results of adaptivity in a spatial setting. By considering collections of Gaussian Markov random fields (GMRF) as approximation sets for the distribution of X, we introduce a novel model selection procedure for spatial fields. For all neighborhoods m in a given collection , this procedure first amounts to computing a covariance estimator of X within the GMRFs of neighborhood m. Then it selects a neighborhood ̂m by applying a penalization strategy. The so-defined method satisfies a nonasymptotic oracle-type inequality. If X is a GMRF, the procedure is also minimax adaptive to the sparsity of its neighborhood. More generally, the procedure is adaptive to the rate of approximation of the true distribution by GMRFs with growing neighborhoods."

Understanding phase transitions probabilistically, as places where the failure of mixing makes the ordinary central limit theorem break down, and non-Gaussian, heavy-tailed distributions appear for macroscopic averages. (I think I bookmarked this in 2000 and then forgot about it... and making me find it again is the only good thing about refereeing this **** paper, grumble.)

I wonder if these guys shouldn't have been at Columbia last week, rather than me...

Recommended-but-not-required text for my class.