Abstract
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 minimum values of the one-point specification of the Markov random field. 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.
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References
Braitenberg, V., Schütz, A.: Cortex: Statistics and Geometry of the Neuronal Connectivity. Springer, Berlin (1998)
Cessac, B.: Statistics of spike trains in conductance-based neural networks: rigorous results. arXiv:1104.3795v2 (2011)
Collet, P., Leonardi, F.: Loss of memory of random functions of Markov chains and Lyapunov exponents. arXiv:0908.0077 (2009)
Dereudre, D., Drouilhet, R., Georgii, H.O.: Existence of Gibbsian point processes with geometry-dependent interactions. Probab. Theory Relat. Fields. 155, 1–28 (2011)
Dobrushin, R.L.: Prescribing a system of random variables by conditional distributions. Theory Probab. Appl. 15, 458–486 (1970)
Georgii, H.O.: Gibbs Measures and Phase Transitions. de Gruyter, Berlin (1988)
Grimmett, G.R.: Percolation. Springer, Berlin (1999)
Liggett, T.M., Schonmann, R.H., Stacey, A.M.: Domination by product measures. Ann. Probab. 25, 71–95 (1997)
Löcherbach, E., Orlandi, V.: Neighborhood radius estimation in variable-neighborhood random fields. Stoch. Process. Appl. 121, 2151–2185 (2011)
MacLean, J., Watson, B., Aaron, G., Yuste, R.: Internal dynamics determine the cortical response to thalamic stimulation. Neuron 48, 811–823 (2005)
Presutti, E.: Scaling limits in statistical mechanics and microstructures in continuum mechanics. Theoretical and Mathematical Physics. Springer, Berlin–Heidelberg (2009)
Rissanen, J.: A universal data compression system. IEEE Trans. Inf. Theory 29, 656–664 (1983)
Russo, L.: The infinite cluster method in the two-dimensional Ising model. Commun. Math. Phys. 67, 251–266 (1979)
Wang, L., Yu, C., Chen, H., Qin, W., He, Y., Fan, F., Zhang, Y., Wang, M., Li, K., Zang, Y., Woodward, T.S., Zhu, C.: Dynamic functional reorganization of the motor execution network after stroke. Brain 133, 1224–1238 (2010)
Acknowledgements
We thank two anonymous referees whose remarks helped us to significantly improve the manuscript. We thank D.Y. Takahashi and R. Fernández for stimulating discussions and bibliographic suggestions. This work is part of USP project MaCLinC, “Mathematics, computation, language and the brain”, USP/COFECUB project “Stochastic systems with interactions of variable range” and CNPq project 476501/2009-1. It was partially supported by CAPES grant AUXPE-PAE-598/2011. A.G. is partially supported by a CNPq fellowship (grant 305447/2008-4). E.L. has been supported by ANR-08-BLAN-0220-01. M.C. and E.L. thank NUMEC, University of Sao Paulo, for hospitality and support.
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Cassandro, M., Galves, A. & Löcherbach, E. Partially Observed Markov Random Fields Are Variable Neighborhood Random Fields. J Stat Phys 147, 795–807 (2012). https://doi.org/10.1007/s10955-012-0488-8
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DOI: https://doi.org/10.1007/s10955-012-0488-8