## Random Fields

*02 Aug 2019 10:05*

Stochastic processes where the index variable is space, or something space-like. (Formally, one-dimensional space works a lot like time; 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 [Mini-review]
- 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:
- 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.]
- 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.]
- Alexei Andreanov, Giulio Biroli, Jean-Philippe Bouchaud, and
Alexandre Lefèvre, "Field theories and exact stochastic equations for
interacting particle
systems", Physical
Review E
**74**(2006): 030101 = cond-mat/0602307 - 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 [JSTOR] - 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 [PS.gz preprint] - Control of Spatially Structured Random Processes and Random Fields with Applications

- "Controlled
Markov Fields with Finite State Space on
Graphs", Stochastic
Models
- 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 [i.e., large deviations]
- 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 - 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
- Mark Kaiser, "Statistical Dependence in Markov Random Field Models" [abstract, preprint]
- 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 ["We give an overview of the ideas central to some recent developments in the ergodic theory of the stochastically forced Navier Stokes equations and other dissipative stochastic partial differential equations."]
- 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
- Jeffrey E. Steif, "Consistent estimation of joint distributions for sufficiently mixing random fields", Annals of
Statistics
**25**(1997): 293--304 [Extension of the Marton-Shields result to random fields in higher dimensions] - 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