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作者：Diaconis, Persi; Lebeau, Gilles; Michel, Laurent
作者单位：Universite Cote d'Azur; Stanford University; Stanford University
摘要：This paper gives geometric tools: comparison, Nash and Sobolev inequalities for pieces of the relevant Markov operators, that give useful bounds on rates of convergence for the Metropolis algorithm. As an example, we treat the random placement of N hard discs in the unit square, the original application of the Metropolis algorithm.
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作者：Cheung, Yitwah; Hubert, Pascal; Masur, Howard
作者单位：California State University System; San Francisco State University; Aix-Marseille Universite; University of Chicago
摘要：Given an irrational 0 < lambda < 1, we consider billiards in the table P-lambda formed by a 1/2 x 1 rectangle with a horizontal barrier of length 1-lambda/2 with one end touching at the midpoint of a vertical side. Let NE(P-lambda) be the set of theta such that the flow on P-lambda in direction theta is not ergodic. We show that the Hausdorff dimension of NE(P-lambda) can only take on the values 0 and 1/2, depending on the summability of the series Sigma(k) log log(qk+1)/q(k) where {q(k)} is t...
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作者：Duplantier, Bertrand; Sheffield, Scott
作者单位：CEA; Centre National de la Recherche Scientifique (CNRS); Universite Paris Saclay; Massachusetts Institute of Technology (MIT)
摘要：Consider a bounded planar domain D, an instance h of the Gaussian free field on D, with Dirichlet energy (2 pi)(-1) integral(D) del h(z) . del h(z)dz, and a constant 0 <= gamma < 2. The Liouville quantum gravity measure on D is the weak limit as epsilon -> 0 of the measures epsilon(gamma 2/2)e(gamma h epsilon(z)) dz, where dz is Lebesgue measure on D and h(epsilon)(z) denotes the mean value of h on the circle of radius epsilon centered at z. Given a random (or deterministic) subset X of D one ...
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作者：Le Rousseau, Jerome; Robbiano, Luc
作者单位：Universite de Orleans; Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI); Universite Paris Saclay; Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI)
摘要：In (0, T) x Omega, Omega open subset of R-n, n >= 2, we consider a parabolic operator P = partial derivative(t) - del(x)delta(t, x)del(x), where the ( scalar) coefficient delta(t, x) is piecewise smooth in space yet discontinuous across a smooth interface S. We prove a global in time, local in space Carleman estimate for P in the neighborhood of any point of the interface. The observation region can be chosen independently of the sign of the jump of the coefficient d at the considered point. T...
