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作者：Hoi Nguyen; Tao, Terence; Van Vu
作者单位：University System of Ohio; Ohio State University; University of California System; University of California Los Angeles; Yale University
摘要：Gaps (or spacings) between consecutive eigenvalues are a central topic in random matrix theory. The goal of this paper is to study the tail distribution of these gaps in various random matrix models. We give the first repulsion bound for random matrices with discrete entries and the first super-polynomial bound on the probability that a random graph has simple spectrum, along with several applications.
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作者：Pinsky, Ross G.; Travers, Nicholas F.
作者单位：Technion Israel Institute of Technology; Indiana University System; Indiana University Bloomington
摘要：We study a random walk on which evolves in a dynamic environment determined by its own trajectory. Sites flip back and forth between two modes, p and q. R consecutive right jumps from a site in the q-mode are required to switch it to the p-mode, and L consecutive left jumps from a site in the p-mode are required to switch it to the q-mode. From a site in the p-mode the walk jumps right with probability p and left with probability , while from a site in the q-mode these probabilities are q and ...
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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：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：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...
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作者：Gueneysu, B.; Matte, O.; Moller, J. S.
作者单位：Humboldt University of Berlin; Aarhus University
摘要：We discuss Hilbert space-valued stochastic differential equations associated with the heat semi-groups of the standard model of non-relativistic quantum electrodynamics and of corresponding fiber Hamiltonians for translation invariant systems. In particular, we prove the existence of a stochastic flow satisfying the strong Markov property and the Feller property. To this end we employ an explicit solution ansatz. In the matrix-valued case, i.e., if the electron spin is taken into account, it i...
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作者：Neuman, Eyal; Zheng, Xinghua
作者单位：University of Rochester; Hong Kong University of Science & Technology
摘要：We study the maximal displacement of a one dimensional subcritical branching random walk initiated by a single particle at the origin. For each let be the rightmost position reached by the branching random walk up to generation n. Under the assumption that the offspring distribution has a finite third moment and the jump distribution has mean zero and a finite probability generating function, we show that there exists such that the function
