Random Fields

Last update: 30 Apr 2026 15:41
First version: Separated from spatial statistics no later than 14 September 2013

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Stochastic processes where the index variable is space, or something space-like. (Formally, one-dimensional space works a lot like time, though not exactly like; and space-time works a lot like a higher-dimensional space, though not always.) This is of course important for modeling spatial and spatio-temporal data, but also data on networks, and statistical mechanics.

See also:
• Cellular Automata;
• Interacting Particle Systems

Recommended, general:
• Pierre Brémaud, Markov Chains: Gibbs Fields, Monte Carlo Simulation, and Queues
• Carlo Gaetan and Xavier Guyon, Spatial Statistics and Modeling
• Geoffrey Grimmett, Probability on Graphs: Random Processes on Graphs and Lattices
• Xavier Guyon, Random Fields on a Network
• Peter Guttorp, Stochastic Modeling of Scientific Data
• Brian D. Ripley, Statistical Inference for Spatial Processes
• Rinaldo B. Schinazi, Classical and Spatial Stochastic Processes

Recommended, of more specialized interest:
• Ole E. Barndorff-Nielsen, Fred Espen Benth and Almut E. D. Veraart, Ambit Stochastics
• J.-R. Chazottes, P. Collet, C. Kuelske and F. Redig, "Deviation inequalities via coupling for stochastic processes and random fields", math.PR/0503483
• Jérôme Dedecker, Paul Doukhan, Gabriel Lang, José Rafael León R., Sana Louhichi and Clémentine Prieur, Weak Dependence: With Examples and Applications
• David Griffeath, "Introduction to Markov Random Fields", ch. 12 in Kemeny, Knapp and Snell, Denumerable Markov Chains [One of the proofs of the equivalence between the Markov property and having a Gibbs distribution, conventionally but misleadingly called the Hammersley-Clifford Theorem. Pollard, below, provides an on-line summary.]
• Mark Kaiser, "Statistical Dependence in Markov Random Field Models" [abstract, preprint]
• Andee Kaplan, Mark S. Kaiser, Soumendra N. Lahiri, Daniel J. Nordman, "Simulating Markov random fields with a conclique-based Gibbs sampler", arxiv:1808.04739 [Presentation by Dr. Kaplan]
• David Pollard, "Markov random fields and Gibbs distributions" [Online PDF. A proof of the theorem linking Markov random fields to Gibbs distributions, following the approach of David Griffeath.]

To read:
• Jan Ambjorn et al., Quantum Geometry: A Statistical Field Theory Approach [I am interested in the stuff about random surfaces.]
• K. Bahlali, M. Eddahbi and M. Mellouk, "Stability and genericity for SPDEs driven by spatially correlated noise", math.PR/0610174
• Raluca M. Balan, "A strong invariance principle for associated random fields", Annals of Probability 33 (2005): 823--840 = math.OR/0503661
• M. S. Bartlett, "Physical Nearest-Neighbour Models and Non-Linear Time Series", Journal of Applied Probability 8 (1971): 222--232
• Michel Bauer, Denis Bernard, "2D growth processes: SLE and Loewner chains", math-ph/0602049
• Denis Belomestny, Vladimir Spokoiny, "Concentration inequalities for smooth random fields", arxiv:1307.1565
• Anton Bovier, Statistical Mechanics of Disordered Systems
• Alexander Bulinski and Alexey Shashkin, "Strong invariance principle for dependent random fields", math.PR/0608237
• M. Cassandro, A. Galves and E. Löcherbach, "Partially Observed Markov Random Fields Are Variable Neighborhood Random Fields", Journal of Statistical Physics 147 (2012): 795--807, arxiv:1111.1177
• Ruslan K. Chornei, Hans Daduna, and Pavel S. Knopov
  • "Controlled Markov Fields with Finite State Space on Graphs", Stochastic Models 21 (2005): 847--874
  • Control of Spatially Structured Random Processes and Random Fields with Applications
• Giuseppe Da Prato, Arnaud Debussche and Luciano Tubaro, "Coupling for some partial differential equations driven by white noise", math.AP/0410441
• Jean-Dominique Deuschel and Andreas Greven (eds.), Interacting Stochastic Systems [This looks deeply cool]
• Rick Durrett, Stochastic Spatial Models: A Hyper-Tutorial
• Vlad Elgart and Alex Kamenev, "Rare Events Statistics in Reaction--Diffusion Systems", cond-mat/0404241
• Mohamed El Machkouri, Dalibor Volny, Wei Biao Wu, "A central limit theorem for stationary random fields", arxiv:1109.0838
• H. Follmer, "On entropy and information gain in random fields", Z. Wahrsh. verw. Geb. 26 (1973): 207--217
• T. Funaki, D. Surgailis and W. A. Woyczynski, "Gibbs-Cox Random Fields and Burgers Turbulence", Annals of Applied Probability 5 (1995): 461--492
• L. Garcia-Ojalvo and J. Sancho, Noise in Spatially Extended Systems
• B. M. Gurevich and A. A. Tempelman, "Markov approximation of homogeneous lattice random fields", Probability Theory and Related Fields 131 (2005): 519--527
• Allan Gut and Ulrich Stadtmuller, "Cesaro Summation for Random Fields", Journal of Theoretical Probability 23 (2010): 715--728
• Peter Hall, Introduction to the Theory of Coverage Processes [= point process with a random shape attached to each point]
• Reza Hosseini, "Conditional information and definition of neighbor in categorical random fields", arxiv:1101.0255 ["Who then is my neighbor?" (Not an actual quote from the paper.)]
• Xiangping Hu, Daniel Simpson, Finn Lindgren, Havard Rue, "Multivariate Gaussian Random Fields Using Systems of Stochastic Partial Differential Equations", arxiv:1307.1379
• Niels Jacob and Alexander Potrykus, "Some thoughts on multiparameter stochastic processes", math.PR/0607744
• Wolfgang Karcher, Elena Shmileva, Evgeny Spodarev, "Extrapolation of stable random fields", arxiv:1107.1654
• M. Kerscher, "Constructing, characterizing, and simulating Gaussian and higher-order point distributions," astro-ph/0102153
• Ross Kindermann and J. Laurie Snell, Markov Random Fields and Their Applications [Free online!]
• P. Kotelenez, Stochastic Space-Time Models and Limit Theorems
• Michael A. Kouritzin and Hongwei Long, "Convergence of Markov chain approximations to stochastic reaction-diffusion equations", Annals of Applied Probability 12 (2002): 1039--1070
• Jean-Francois Le Gall, Spatial Branching Processes, Random Snakes and Partial Differential Equations
• Atul Mallik, Michael Woodroofe, "A Central Limit Theorem For Linear Random Fields", arxiv:1007.1490
• Jonathan C. Mattingly, "On Recent Progress for the Stochastic Navier Stokes Equations", math.PR/0409194
• A. I. Olemskoi, D. O. Kahrchenko and I. A. Knyaz', "Phase transitions induced by noise cross-correlations", cond-mat/0403583
• Rupert Paget, "Strong Markov Random Field Model", IEEE Transactions on Pattern Analysis and Machine Intelligence 26 (2004): 408--413
• Marcelo Pereyra, Nicolas Dobigeon, Hadj Batatia, Jean-Yves Tourneret, "Computing the Cramer-Rao bound of Markov random field parameters: Application to the Ising and the Potts models", arxiv:1206.3985
• Liang Qiao, Radek Erban, C. T. Kelley and Ioannis G. Kevrekidis, "Spatially Distributed Stochastic Systems: equation-free and equation-assisted preconditioned computation", q-bio.QM/0606006
• Havard Rue and Leonhard Held, Gaussian Markov Random Fields: Theory and Applications
• Andre Toom, "Law of Large Numbers for Non-Local Functions of Probabilistic Cellular Automata", Journal of Statistical Physics 133 (2008): 883--897
• M. N. M. van Lieshout, "Markovianity in space and time", math.PR/0608242
• Divyanshu Vats and Jose M. F. Moura, "Telescoping Recursive Representations and Estimation of Gauss-Markov Random Fields", arxiv:0907.5397
• Benjamin Yakir, Extremes in Random Fields: A Theory and its Applications
• Eunho Yang, Pradeep K. Ravikumar, Genevera I. Allen, Zhandong Liu, "Conditional Random Fields via Univariate Exponential Families", NIPS 2013