S. Diaconis, Then Shiu and Chen (2015) used this result to study the convergence rate of the Gibbs sampler for the 1?D Ising model. In the first part of this thesis, we generalize the result of Shiu and Chen to the case of the 1?D Potts model with three and more spin states (see Theorem 2.2.1 and Theorem 2.2.2). In the second part we generalize their method to the case of the 2?D Ising model with two spin states (see Theorem 3.2.3), 1991.

, Note that in the case of the 2?D Ising model (see Theorem 3.2.3) we did not succeed to compute the Diaconis and Stroock bound as explicitly in Theorem 3.2.3 as in the one dimensional case. We only obtained an upper bound for the bound they give for ? 1 . This raises the following question: Is there a way to use some symmetry argument, like in the one dimensional case, to obtain an exact expression for the Diaconis Stroock bound?, We study in the second and third chapter of this thesis the rate of convergence of the Gibbs sampler algorithm for some model from statistical mechanics

M. , The difficulty is to understand the Cheeger constant at finite temperature for Ising model and Potts model. Yet another method is to use coupling to bound the convergence time

?. Model, We think that we can use this approach in the case of the Potts model

D. Aldous, Random walks on finite groups and rapidly mixing Markov chain. In séminaire de probabilités XVII 1981/82, Lecture Notes in Mathematics, vol.986, pp.243-97, 1983.

C. Bandle, Isoperimetric Inequalities and Applications, 1980.

T. Chen, W. Chen, C. Hwang, and H. Pai, On the optimal transition matrix for Markov chain Monte Carlo sampling, J. Control Optim, vol.59, pp.2743-2762, 2012.

T. Chen and C. Hwang, Accelerating reversible Markov chains, Statist. Probab. Lett, vol.83, issue.9, pp.1956-1962, 2013.

P. Diaconis and D. Stroock, Geometric bounds for eigenvalues of Markov chains, Ann. Appl. Probab, vol.1, issue.1, pp.36-61, 1991.

R. Durrett, Probability: Theory and examples, 1996.

B. Franke and A. Helali, The Convergence rate of the Gibbs sampler for the 2-D Ising model via a geometric bound, 2018.

B. Franke and A. Helali, On the convergence rate of some perturbed Gibbs sampler for the 1?D Ising model, 2018.

A. Frigessi, C. Hwang, S. Sheu, and P. Di-stefano, Convergence rate of the Gibbs sampler, the Metropolis algorithm, and other single site update dynamics, J. R. Stat. Soc. Ser. B, vol.55, pp.205-219, 1993.

A. Frigessi, C. Hwang, and . Younes, Optimal spectral structure of reversible stochastic matrices, Monte Carlo methods and the simulation of Markov random fields, Ann. Appl. Probab, vol.2, pp.610-628, 1992.

S. Geman and D. Geman, Stochastic relaxation, Gibbs distributions and the Bayesian restoration of images, IEEE Trans. Pattern Anal. Mach. Intell, vol.6, pp.721-741, 1984.

A. L. Gibbs, Bounding the convergence time of the Gibbs sampler in Bayesian image restoration, Biometrika, vol.87, issue.4, pp.749-766, 2000.

W. K. Hastings, Monte Carlo sampling methods using Markov chains and their applications, Biometrika, vol.57, pp.97-109, 1970.

A. Helali, The convergence rate of the Gibbs sampler for generalized 1-D Ising model, 2018.

H. Holley and D. Stroock, Simulated Annealing via Sobolev Inequalities, Communications in Mathematical Physics, vol.115, pp.553-569, 1988.

R. Horn and C. Johnson, Matrix Analysis, 1985.

H. Landau and A. Odlyzko, Bounds for eigenvalues of certain stochastic matrices, Linear Algebra Appl, vol.38, pp.5-15, 1981.

M. Luby, D. Randall, and A. Sinclair, Markov chain algorithms for planar Lattice structures (extended abstract), In Symposium on Foundation of Computer Science, vol.36, pp.150-159, 1995.

M. Losifescu, Finite Markov processes and their applications

S. Ingrassia, On the rate of convergence of the metropolis algorithm and Gibbs sampler by geometric bounds, Ann. Appl. Probab, vol.4, pp.347-389, 1994.

N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, and E. Teller, Equations of state calculations by fast computing machines, J. Chem. Phys, vol.21, pp.1087-1092, 1953.

B. Mohar, Eigenvalues, diameter, and mean distance in graphs, Dept. Mathematics, Univ. E. K. Ljubljanos, 1989.

B. Mohar and Y. Jadranska, The Laplacian spectrum of graphs, Dept. Mathematics, Univ. E. K. Ljubljanos, 1989.

J. Propp, G. Wilson, D. , and B. , Exact sampling with coupled Markov chain and Applications to statistical mechanics, Random Stuct. Algor, vol.9, pp.223-52, 1996.

S. Shiu and C. , On the rate of convergence of the Gibbs sampler for the 1-D Ising model by geometric bound, Statistics and Probability Letters, vol.105, pp.14-19, 2015.

A. Sinclair, Improved bounds for mixing rates of Markov chains and multicommodity flow, Comb. Probab. Comput, vol.1, pp.351-370, 1991.

A. Sinclair and M. Jerrum, Approximate counting, uniform generation and rapidly mixing Markov chain, Inform. and Comput, vol.82, pp.93-133, 1989.

G. Winkler, Image Analysis, Random fields and Markov Chain Monte Carlo methods: A mathematical Introduction, vol.27, 2012.