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作者：Le Gall, Jean-Francois; Riera, Armand
作者单位：Universite Paris Cite
摘要：We provide a unified approach to the three main non-compact models of random geometry, namely the Brownian plane, the infinite-volume Brownian disk, and the Brownian half-plane. This approach allows us to investigate relations between these models, and in particular to prove that complements of hulls in the Brownian plane are infinite-volume Brownian disks.
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作者：Miller, Jason; Sheffield, Scott
作者单位：University of Cambridge; Massachusetts Institute of Technology (MIT)
摘要：Previous works in this series have shown that an instance of a root 8/3-Liouville quantum gravity (LQG) sphere has a well-defined distance function, and that the resulting metric measure space (mm-space) agrees in law with the Brownian map (TBM). In this work, we show that given just the mm-space structure, one can a.s. recover the LQG sphere. This implies that there is a canonical way to parameterize an instance of TBM by the Euclidean sphere (up to Mobius transformation). In other words, an ...
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作者：Barlow, M. T.; Croydon, D. A.; Kumagai, T.
作者单位：University of British Columbia; Kyoto University
摘要：This article investigates the heat kernel of the two-dimensional uniform spanning tree. We improve previous work by demonstrating the occurrence of log-logarithmic fluctuations around the leading order polynomial behaviour for the on-diagonal part of the quenched heat kernel. In addition we give two-sided estimates for the averaged heat kernel, and we show that the exponents that appear in the off-diagonal parts of the quenched and averaged versions of the heat kernel differ. Finally, we deriv...
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作者：Duminil-Copin, Hugo; Manolescu, Ioan; Tassion, Vincent
作者单位：University of Geneva; Universite Paris Saclay; University of Fribourg; Swiss Federal Institutes of Technology Domain; ETH Zurich
摘要：This paper is studying the critical regime of the planar random-cluster model on Z(2) with cluster-weight q is an element of [1, 4). More precisely, we prove crossing estimates in quads which are uniform in their boundary conditions and depend only on their extremal lengths. They imply in particular that any fractal boundary is touched by macroscopic clusters, uniformly in its roughness or the configuration on the boundary. Additionally, they imply that any sub-sequential scaling limit of the ...
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作者：Addario-Berry, Louigi; Sen, Sanchayan
作者单位：McGill University; Indian Institute of Science (IISC) - Bangalore
摘要：The global structure of the minimal spanning tree (MST) is expected to be universal for a large class of underlying random discrete structures. However, very little is known about the intrinsic geometry of MSTs of most standard models, and so far the scaling limit of the MST viewed as a metric measure space has only been identified in the case of the complete graph (Addario-Berry et al. in Ann Probab 45(5):3075-3144, 2017). In this work, we show that the MST constructed by assigning i.i.d. con...
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作者：Asselah, Amine; Schapira, Bruno
作者单位：Universite Gustave-Eiffel; Universite Paris-Est-Creteil-Val-de-Marne (UPEC); Centre National de la Recherche Scientifique (CNRS); Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI); Aix-Marseille Universite
摘要：We obtain sharp upper and lower bounds for the downward moderate deviations of the volume of the range of a random walk in dimension five and larger. Our results encompass two regimes: a Gaussian regime for small deviations, and a stretched exponential regime for larger deviations. In the latter regime, we show that conditioned on the moderate deviations event, the walk folds a small part of its range in a ball-like subset. Also, we provide new path properties, in dimension three as well. Besi...
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作者：Lubetzky, Eyal; Sly, Allan
作者单位：New York University; Princeton University
摘要：In the study of Markov chain mixing times, analysis has centered on the performance from a worst-case starting state. Here, in the context of Glauber dynamics for the one-dimensional Ising model, we show how new ideas from information percolation can be used to establish mixing times from other starting states. At high temperatures we show that the alternating initial condition is asymptotically the fastest one, and, surprisingly, its mixing time is faster than at infinite temperature, acceler...
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作者：Baik, Jinho; Liu, Zhipeng
作者单位：University of Michigan System; University of Michigan; University of Kansas
摘要：We consider the one-dimensional totally asymmetric simple exclusion process with an arbitrary initial condition in a spatially periodic domain, and obtain explicit formulas for the multi-point distributions in the space-time plane. The formulas are given in terms of an integral involving a Fredholm determinant. We then evaluate the large-time, large-period limit in the relaxation time scale, which is the scale such that the system size starts to affect the height fluctuations. The limit is obt...
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作者：Bertoin, Jean
作者单位：University of Zurich
摘要：Let X-1, X-2,... be i.i.d. copies of some real random variable X. For any deterministic epsilon(2), epsilon(3),... in {0, 1}, a basic algorithm introduced by H.A. Simon yields a reinforced sequence (X) over cap (1), (X) over cap (2),... as follows. If epsilon(n) = 0, then (X) over cap (n) is a uniform random sample from (X) over cap (1), ... , (X) over cap (n-1); otherwise (X) over cap (n) is a newindependent copy of X. The purpose of this work is to compare the scaling exponent of the usual r...
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作者：Barraquand, Guillaume; Corwin, Ivan; Dimitrov, Evgeni
作者单位：Universite PSL; Ecole Normale Superieure (ENS); Universite Paris Cite; Sorbonne Universite; Centre National de la Recherche Scientifique (CNRS); Columbia University
摘要：We prove that the free energy of the log-gamma polymer between lattice points (1, 1) and (M, N) converges to the GUE Tracy-Widom distribution in the M-1/3 scaling, provided that N/M remains bounded away from zero and infinity. We prove this result for the model with inverse gamma weights of any shape parameter theta > 0 and furthermore establish a moderate deviation estimate for the upper tail of the free energy in this case. Finally, we consider a non i.i.d. setting where the weights on finit...
