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作者：Huang, Jiaoyang; Landon, Benjamin; Yau, Horng-Tzer
作者单位：Harvard University
摘要：We consider the statistics of the extreme eigenvalues of sparse random matrices, a class of random matrices that includes the normalized adjacency matrices of the Erdos-Renyi graph G(N, p). Tracy-Widom fluctuations of the extreme eigenvalues for p >> N-2/3 was proved in (Probab. Theory Related Fields 171 (2018) 543-616; Comm. Math. Phys. 314 (2012) 587-640). We prove that there is a crossover in the behavior of the extreme eigenvalues at p similar to N-2/3. In the case that N-7/9 << p << N-2/3...
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作者：Dembin, Barbara
作者单位：Universite Paris Cite
摘要：We consider the standard first passage percolation model on Z(d) with a distribution G on R+ that admits an exponential moment. We study the maximal flow between a compact convex subset A of R-d and infinity. The study of maximal flow is associated with the study of sets of edges of minimal capacity that cut A from infinity. We prove that the rescaled maximal flow between nA and infinity phi(nA)/n(d-1) almost surely converges toward a deterministic constant depending on A. This constant corres...
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作者：Janjigian, Christopher; Rassoul-Agha, Firas
作者单位：Utah System of Higher Education; University of Utah
摘要：We consider random walk in a space-time random potential, also known as directed random polymer measures, on the planar square lattice with nearest-neighbor steps and general i.i.d. weights on the vertices. We construct covariant cocycles and use them to prove new results on existence, uniqueness/nonuniqueness, and asymptotic directions of semi-infinite polymer measures (solutions to the Dobrushin-Lanford-Ruelle equations). We also prove nonexistence of covariant or deterministically directed ...
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作者：Alt, Johannes; Erdos, Laszlo; Krueger, Torben; Schroeder, Dominik
作者单位：University of Geneva; Institute of Science & Technology - Austria; University of Bonn
摘要：We prove edge universality for a general class of correlated real symmetric or complex Hermitian Wigner matrices with arbitrary expectation. Our theorem also applies to internal edges of the self-consistent density of states. In particular, we establish a strong form of band rigidity which excludes mismatches between location and label of eigenvalues close to internal edges in these general models.
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作者：Chatterjee, Sourav; Dunlap, Alexander
作者单位：Stanford University; Stanford University
摘要：The (d + 1)-dimensional KPZ equation is the canonical model for the growth of rough d-dimensional random surfaces. A deep mathematical understanding of the KPZ equation for d = 1 has been achieved in recent years, and the case d >= 3 has also seen some progress. The most physically relevant case of d = 2, however, is not very well understood mathematically, largely due to the renormalization that is required: in the language of renormalization group analysis, the d = 2 case is neither ultravio...
