We study the asymptotic behavior of Masur-Veech volumes as the genus goes to infinity. We show the existence of a complete asymptotic expansion of these volumes that depends only on the genus and the number of singularities. The computation of the first term of this asymptotics expansion was a long standing problem. This problem was recently solved in by Aggarwal using purely combinatorial arguments, and then by D. Chen, M. Moeller, D. Zagier and the author using algebro-geometric insights. Our proof relies on a combination of both methods.
These problems often admits a counterpart in terms of asymptotic dynamical properties of random surfaces (see @cite_3 or @cite_7 ), or in string theory: one can either consider systems with a large number of particles (see @cite_4 ) or the asymptotic behavior of perturbative expansions (see @cite_5 ).
@cite_5 @cite_4 @cite_7 @cite_3
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We study the asymptotic statistical behavior of the 2-dimensional periodic Lorentz gas with an infinite horizon. We consider a particle moving freely in the plane with elastic reflections from a periodic set of fixed convex scatterers. We assume that the initial position of the particle in the phase space is random with uniform distribution with respect to the Liouville measure of the periodic problem. We are interested in the asymptotic statistical behavior of the particle displacement in the plane as the timet goes to infinity. We assume that the particle horizon is infinite, which means that the length of free motion of the particle is unbounded. Then we show that under some natural assumptions on the free motion vector autocorrelation function, the limit distribution of the particle displacement in the plane is Gaussian, but the normalization factor is (t logt)1 2 and nott1 2 as in the classical case. We find the covariance matrix of the limit distribution. We perform simulations for one dimensional continuous-time random walks in two dynamic random environments with fast (independent spin-flips) and slow (simple symmetric exclusion) decay of space-time correlations, respectively. We focus on the asymptotic speeds and the scaling limits of such random walks. We observe different behaviors depending on the dynamics of the underlying random environment and the ratio between the jump rate of the random walk and the one of the environment. We compare our data with well known results for static random environment. We observe that the non-diffusive regime known so far only for the static case can occur in the dynamical setup too. Such anomalous fluctuations give rise to a new phase diagram. Further we discuss possible consequences for more general static and dynamic random environments. This work is a follow-up to our previous work "A numerical approach related to defect-type theories for some weakly random problems in homogenization" (preprint available on this archive). It extends and complements, both theoretically and experimentally, the results presented there. Under consideration is the homogenization of a model of a weakly random heterogeneous material. The material consists of a reference periodic material randomly perturbed by another periodic material, so that its homogenized behavior is close to that of the reference material. We consider laws for the random perturbations more general than in our previous work cited above. We prove the validity of an asymptotic expansion in a certain class of settings. We also extend the formal approach introduced in our former work. Our perturbative approach shares common features with a defect-type theory of solid state physics. The computational efficiency of the approach is demonstrated. This paper studies a class of non-Markovian and nonhomogeneous stochastic processes on a finite state space. Relying on a recent paper by Benaim, Hofbauer, and Sorin [SIAM J. Control Optim., 44 (2005), pp. 328-348] it is shown that, under certain assumptions, the asymptotic behavior of occupation measures can be described in terms of a certain set-valued deterministic dynamical system. This provides a unified approach to simulated annealing type processes and permits the study of new models of vertex reinforced random walks and new models of learning in games such as Markovian fictitious play.