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作者：Basu, Riddhipratim; Bhatia, Manan; Ganguly, Shirshendu
作者单位：Tata Institute of Fundamental Research (TIFR); University of California System; University of California Berkeley
摘要：First passage percolation (FPP) on Z d or R d is a canonical model of a random metric space where the standard Euclidean geometry is distorted by random noise. Of central interest is the length and the geometry of the geodesic, the shortest path between points. Since the latter, owing to its length minimization, traverses through atypically low values of the underlying noise variables, it is an important problem to quantify the disparity between the environment rooted at a point on the geodesi...
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作者：Bauerschmidt, Roland; Park, Jiwoon; Rodriguez, Pierre -Francois
作者单位：University of Cambridge; Imperial College London
摘要：The Discrete Gaussian model is the lattice Gaussian free field conditioned to be integer -valued. In two dimensions and at sufficiently high temperature, we show that its macroscopic scaling limit on the torus is a multiple of the Gaussian free field. Our proof starts from a single renormalisation group step after which the integer -valued field becomes a smooth field, which we then analyse using the renormalisation group method. This paper also provides the foundation for the construction of ...
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作者：Iksanov, Alexander; Kolesko, Konrad; Meiners, Matthias
作者单位：Ministry of Education & Science of Ukraine; Taras Shevchenko National University of Kyiv; University of Wroclaw; Justus Liebig University Giessen
摘要：Consider a supercritical Crump-Mode-Jagers process (Z(t)(phi))(t >= 0) counted with a random characteristic phi. Nerman's celebrated law of large numbers [Z. Wahrsch. Verw. Gebiete 57, 365--395, 1981] states that, under some mild assumptions, e(-alpha t)Z(t )(phi)converges almost surely as t ->infinity to aW. Here, alpha>0 is the Malthusian parameter, a is a constant and W is the limit of Nerman's martingale, which is positive on the survival event. In this general situation, under additional ...
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作者：Bauerschmidt, Roland; Park, Jiwoon; Rodriguez, Pierre-Francois
作者单位：University of Cambridge; Imperial College London
摘要：The Discrete Gaussian model is the lattice Gaussian free field conditioned to be integer-valued. In two dimensions and at sufficiently high temperature, we show that the scaling limit of the infinite-volume gradient Gibbs state with zero mean is a multiple of the Gaussian free field. This article is the second in a series on the Discrete Gaussian model, extending the methods of the first paper by the analysis of general external fields (rather than macroscopic test functions on the torus). As ...
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作者：Chapuy, Guillaume; Louf, Baptiste; Walsh, Harriet
作者单位：Universite Paris Cite; Centre National de la Recherche Scientifique (CNRS); Uppsala University; Centre National de la Recherche Scientifique (CNRS); Universite Paris Cite; Ecole Normale Superieure de Lyon (ENS de LYON)
摘要：We study the asymptotic behaviour of random integer partitions under a new probability law that we introduce, the Plancherel-Hurwitz measure. This distribution, which has a natural definition in terms of Young tableaux, is a deformation of the classical Plancherel measure, which appears naturally in the context of Hurwitz numbers, enumerating certain transposition factorisations in symmetric groups. We study a regime in which the number of factors in the underlying factorisations grows linearl...
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作者：Gorski, Christian
作者单位：Northwestern University
摘要：We prove a strict monotonicity theorem for first passage percolation on any Cayley graph of a virtually nilpotent group which is not isomorphic to the standard Cayley graph of Z: given two distributions v and v with finite mean, if v is strictly more variable than v and v is subcritical in an appropriate sense, then the expected passage times associated to v exceed those of v by an amount proportional to the graph distance. This generalizes a theorem of van den Berg and Kesten from 1993, which...
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作者：Tikhomirov, Konstantin; Youssef, Pierre
作者单位：University System of Georgia; Georgia Institute of Technology; New York University; New York University Abu Dhabi
摘要：In this work we develop a comparison procedure for the modified logfinite state space. Efficient comparison of the MLSI Dirichlet forms is a wellknown obstacle in the theory of Markov chains. We approach this problem by introducing a regularized MLSI constant, which, under some assumptions, has the same order of magnitude as the usual MLSI constant yet is amenable for comparison and thus considerably simpler to estimate in certain cases. As an application of this general comparison procedure, ...
