Università Roma Tre

1. Random walks and Markov chains Successions of random variables. Random walks. Discrete and continuous time Markov chains. Invariant measurement, time-reversal and reversibility a2. Examples and classic models. Random walks on graphs. Birth and death processes. Exclusion processes. Monte Carlo method: Glauber's dynamic Metropolise type algorithms for Ising modeling, coloring of a graph and other interacting systems. 3. Convergence to equilibrium I. Distance in variation, mixing times. Teoremiergodici. Coupling techniques. Strong stationary times. Applications to the coupon collector problem and to the shuffling of a deck of cards.4. Convergence to equilibrium II. Convergence in norm L2. Spectral gap and estimated relaxation times. Cheeger inequality, conductance and cam method. "Comparison" method. Spectral gap for the d-dimensional sultore exclusion process. Convergence to equilibrium in terms of logarithmic entropy and inequality. Esempi.5. Other topics chosen. Glauber's dynamics for the Ising model: dynamic phase transition for the medium field model and for the suZ2 model. The "cut-off" phenomenon. Logarithmic Sobolev inequalities and equilibrium convergence. Algorithm for the "perfect simulation".

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