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作者：Stauffer, Alexandre
作者单位：University of Bath
摘要：We study random triangulations of the integer points [0, n](2) boolean AND Z(2), where each triangulation has probability measure with being a real parameter and denoting the sum of the length of the edges in . Such triangulations are called lattice triangulations. We construct a height function on lattice triangulations and prove that, in the whole subcritical regime , the function behaves as a Lyapunov function with respect to Glauber dynamics; that is, the function is a supermartingale. We ...
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作者：Bettinelli, Jeremie; Miermont, Gregory
作者单位：Centre National de la Recherche Scientifique (CNRS); Institut Polytechnique de Paris; Ecole Polytechnique; Institut Polytechnique de Paris; Ecole Polytechnique; Ecole Normale Superieure de Lyon (ENS de LYON); Institut Universitaire de France
摘要：We show that, under certain natural assumptions, large random plane bipartite maps with a boundary converge after rescaling to a one-parameter family of random metric spaces homeomorphic to the closed unit disk of , the space being called the Brownian disk of perimeter L and unit area. These results can be seen as an extension of the convergence of uniform plane quadrangulations to the Brownian map, which intuitively corresponds to the limit case where . Similar results are obtained for maps f...
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作者：Georgiou, Nicos; Rassoul-Agha, Firas; Seppalainen, Timo
作者单位：University of Sussex; Utah System of Higher Education; University of Utah; University of Wisconsin System; University of Wisconsin Madison
摘要：We study the directed last-passage percolation model on the planar square lattice with nearest-neighbor steps and general i.i.d. weights on the vertices, outside of the class of exactly solvable models. Stationary cocycles are constructed for this percolation model from queueing fixed points. These cocycles serve as boundary conditions for stationary last-passage percolation, solve variational formulas that characterize limit shapes, and yield existence of Busemann functions in directions wher...
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作者：Bao, Zhigang; Erdos, Laszlo
作者单位：Institute of Science & Technology - Austria
摘要：We consider Hermitian random matrices H consisting of blocks of size . The matrix elements are i.i.d. within the blocks, close to a Gaussian in the four moment matching sense, but their distribution varies from block to block to form a block-band structure, with an essential band width M. We show that the entries of the Green's function satisfy the local semicircle law with spectral parameter down to the real axis for any , using a combination of the supersymmetry method inspired by Shcherbina...
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作者：Soo, Terry
作者单位：University of Kansas
摘要：In the simple case of a Bernoulli shift on two symbols, zero and one, by permuting the symbols, it is obvious that any two equal entropy shifts are isomorphic. We show that the isomorphism can be realized by a factor that maps a binary sequence to another that is coordinatewise smaller than or equal to the original sequence.
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作者：Miller, Jason; Sheffield, Scott
作者单位：Microsoft; Massachusetts Institute of Technology (MIT); University of Cambridge
摘要：We establish existence and uniqueness for Gaussian free field flow lines started at interior points of a planar domain. We interpret these as rays of a random geometry with imaginary curvature and describe the way distinct rays intersect each other and the boundary. Previous works in this series treat rays started at boundary points and use Gaussian free field machinery to determine which chordal processes are time-reversible when . Here we extend these results to whole-plane and establish con...
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作者：Ganguly, Shirshendu; Levine, Lionel; Peres, Yuval; Propp, James
作者单位：University of Washington; University of Washington Seattle; Cornell University; Microsoft; University of Massachusetts System; University of Massachusetts Lowell
摘要：We introduce a graph-theoretic model of interface dynamics called competitive erosion. Each vertex of the graph is occupied by a particle that can be either red or blue. New red and blue particles alternately get emitted from their respective bases and perform random walk. On encountering a particle of the opposite color they kill it and occupy its position. We prove that on the cylinder graph (the product of a path and a cycle) an interface spontaneously forms between red and blue and is main...
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作者：Kosygina, Elena; Zerner, Martin P. W.
作者单位：City University of New York (CUNY) System; Baruch College (CUNY); Eberhard Karls University of Tubingen
摘要：We provide sufficient conditions for the validity of a dichotomy, i.e. zero-one law, between recurrence and transience of general frog models. In particular, the results cover frog models with i.i.d. numbers of frogs per site where the frog dynamics are given by quasi-transitive Markov chains or by random walks in a common random environment including super-critical percolation clusters on . We also give a sufficient and almost sharp condition for recurrence of uniformly elliptic frog processe...
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作者：Jagannath, Aukosh; Tobasco, Ian
摘要：In this paper we study the Parisi variational problem for mixed p-spin glasses with Ising spins. Our starting point is a characterization of Parisi measures whose origin lies in the first order optimality conditions for the Parisi functional, which is known to be strictly convex. Using this characterization, we study the phase diagram in the temperature-external field plane. We begin by deriving self-consistency conditions for Parisi measures that generalize those of de Almeida and Thouless to...
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作者：Basak, Anirban; Mukherjee, Sumit
作者单位：Duke University; Columbia University
摘要：We consider the Potts model with q colors on a sequence of weighted graphs with adjacency matrices , allowing for both positive and negative weights. Under a mild regularity condition on we show that the mean-field prediction for the log partition function is asymptotically correct, whenever . In particular, our results are applicable for the Ising and the Potts models on any sequence of graphs with average degree going to . Using this, we establish the universality of the limiting log partiti...
