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Probability and Geometry on Groups  Lecture notes for a graduate course
, 2015
"... These notes have grown (and are still growing) out of two graduate courses I gave at the University of Toronto. The main goal is to give a selfcontained introduction to several interrelated topics of current research interests: the connections between 1) coarse geometric properties of Cayley grap ..."
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These notes have grown (and are still growing) out of two graduate courses I gave at the University of Toronto. The main goal is to give a selfcontained introduction to several interrelated topics of current research interests: the connections between 1) coarse geometric properties of Cayley graphs of infinite groups; 2) the algebraic properties of these groups; and 3) the behaviour of probabilistic processes (most importantly, random walks, harmonic functions, and percolation) on these Cayley graphs. I try to be as little abstract as possible, emphasizing examples rather than presenting theorems in their most general forms. I also try to provide guidance to recent research literature. In particular, there are presently over 150 exercises and many open problems that might be accessible to PhD students. It is also hoped that researchers working either in probability or in geometric group theory will find these notes useful to enter the other field.
Glauber dynamics on nonamenable graphs: boundary conditions and mixing time
, 2008
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Reversibility and mixing time for logit dynamics with concurrent updates. CoRR abs/1207.2908
"... Logit dynamics [Blume, Games and Economic Behavior, 1993] is a randomized best response dynamics whereat everytime step a playeris selected uniformly at randomand she choosesanewstrategyaccordingtothe“logitchoicefunction”, i.e. aprobabilitydistribution biased towards strategies promising higher payo ..."
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Logit dynamics [Blume, Games and Economic Behavior, 1993] is a randomized best response dynamics whereat everytime step a playeris selected uniformly at randomand she choosesanewstrategyaccordingtothe“logitchoicefunction”, i.e. aprobabilitydistribution biased towards strategies promising higher payoffs, where the bias level corresponds to the degreeofrationalityoftheagents. Whilethelogitchoicefunctionisaverynaturalbehavioral model for approximately rational agents, the specific revision process that selects one single player per time step seems less justified. In this paper we thus focus on the dynamics where at every time step every player simultaneously updates her strategy according to the logit choice function. We call such a dynamics the “alllogit”, as opposed to the classical “onelogit ” dynamics. ThealllogitdynamicsforagameinducesanergodicMarkovchainoverthesetofstrategy profiles which is significantly different from the Markov chain induced in the onelogit case. In this paper we first highlight similarities and differences between the two dynamics with some simple examples of twoplayer games; we then give a characterization of the class of games such that the Markov chains induced by the alllogit dynamics are reversible and we show it is a subclass of potential games; finally, we analyze the mixing time of the alllogit dynamics for a wellknown coordination game. 1 1
Logit Dynamics with Concurrent Updates for Local Interaction Games?
"... Abstract. Logit dynamics are a family of randomized best response dynamics based on the logit choice function [21] that is used to model players with limited rationality and knowledge. In this paper we study the alllogit dynamics, where at each time step all players concurrently update their strate ..."
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Abstract. Logit dynamics are a family of randomized best response dynamics based on the logit choice function [21] that is used to model players with limited rationality and knowledge. In this paper we study the alllogit dynamics, where at each time step all players concurrently update their strategies according to the logit choice function. In the well studied onelogit dynamics [7] instead at each step only one randomly chosen player is allowed to update. We study properties of the alllogit dynamics in the context of local interaction games, a class of games that has been used to model complex social phenomena [7, 23, 26] and physical systems [19]. In a local interaction game, players are the vertices of a social graph whose edges are twoplayer potential games. Each player picks one strategy to be played for all the games she is involved in and the payoff of the player is the (weighted) sum of the payoffs from each of the games. We prove that local interaction games characterize the class of games for which the alllogit dynamics are reversible. We then compare the stationary behavior of onelogit and alllogit dynamics. Specifically, we look at the expected value of a notable class of observables, that we call decomposable observables. 1
DYNAMICS OF EXPONENTIAL LINEAR MAP IN FUNCTIONAL SPACE
, 2004
"... Abstract. We consider the question of existence of a unique invariant probability distribution which satisfies some evolutionary property. The problem arises from the random graph theory but to answer it we treat it as a dynamical system in the functional space, where we look for a global attractor. ..."
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Abstract. We consider the question of existence of a unique invariant probability distribution which satisfies some evolutionary property. The problem arises from the random graph theory but to answer it we treat it as a dynamical system in the functional space, where we look for a global attractor. We consider the following bifurcation problem: Given a probability measure µ, which corresponds to the weight distribution of a link of a random graph we form a positive linear operator Φ (convolution) on distribution functions and then we analyze a family of its exponents with a parameter λ which corresponds to connectivity of a sparse random graph. We prove that for every measure µ (i.e., convolution Φ) and every λ < e there exists a unique globally attracting fixed point of the operator, which yields the existence and uniqueness of the limit probability distribution on the random graph. This estimate was established earlier [KS81] for deterministic weight distributions (Dirac measures µ) and is known as ecutoff phenomena, as for such distributions and λ> e there is no fixed point attractor. We thus establish this phenomenon in a much more general sense. 1.
GLAUBER DYNAMICS ON HYPERBOLIC GRAPHS: BOUNDARY CONDITIONS AND MIXING TIME
, 2008
"... We study a continuous time Glauber dynamics reversible with respect to the Ising model on hyperbolic graphs and analyze the effect of boundary conditions on the mixing time. Specifically, we consider the dynamics on an nvertex ball of the hyperbolic graph H(v, s), where v is the number of neighbor ..."
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We study a continuous time Glauber dynamics reversible with respect to the Ising model on hyperbolic graphs and analyze the effect of boundary conditions on the mixing time. Specifically, we consider the dynamics on an nvertex ball of the hyperbolic graph H(v, s), where v is the number of neighbors of each vertex and s is the number of sides of each face, conditioned on having (+)boundary. If v> 4, s> 3 and for all low enough temperatures (phase coexistence region) we prove that the spectral gap of this dynamics is bounded below by a constant independent of n. This implies that the mixing time grows at most linearly in n, in contrast to the free boundary case where it is polynomial with exponent growing with the inverse temperature β. Such a result extends to hyperbolic graphs the work done by Martinelli, Sinclair and Weitz for the analogous system on regular tree graphs, and provides a further example of influence of the boundary condition on the mixing time.