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作者：Ji, Hong chang; Park, Jaewhi
作者单位：Institute of Science & Technology - Austria; Korea Advanced Institute of Science & Technology (KAIST)
摘要：We consider fluctuations of the largest eigenvalues of the random matrix model A + UBU* where A and B are N x N deterministic Hermitian (or symmetric) matrices and U is a Haar-distributed unitary (or orthogonal) matrix. We prove that the largest eigenvalue weakly converges to the GUE (or GOE) Tracy-Widom distribution, under mild assumptions on A and B to guarantee that the density of states of the model decays as square root around the upper edge. Our proof is based on the comparison of the Gr...
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作者：Salins, Michael
作者单位：Boston University
摘要：We investigate the finite time explosion of the stochastic heat equation (partial derivative u)/(partial derivative t)=Delta u(t,x)+sigma(u(t,x))W-center dot(t,x) in the critical setting where sigma grows like sigma(u)approximate to C(1+|u|(gamma)) and gamma=(3)/(2). Mueller previously identified gamma=(3)/(2) as the critical growth rate for explosion and proved that solutions cannot explode in finite time if gamma<(3)/(2) and solutions will explode with positive probability if gamma>(3)/(2). ...
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作者：Disertori, Margherita; Merkl, Franz; Rolles, Silke W. W.
作者单位：University of Bonn; University of Bonn; University of Munich; Technical University of Munich
摘要：We show that the vertex-reinforced jump process on the d-dimensional lattice with long-range jumps is transient in any dimension d as long as the initial weights do not decay too fast. The main ingredients in the proof are: an analysis of the corresponding random environment on finite boxes, a comparison with a hierarchical model, and the reduction of the hierarchical model to a nonhomogeneous effective one-dimensional model. For d >= 3 we also prove transience of the vertex-reinforced jump pr...
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作者：Heilman, Steven
作者单位：University of Southern California
摘要：We prove a vector-valued inequality for the Gaussian noise stability (i.e. we prove a vector-valued Borell inequality) for Euclidean functions taking values in the two-dimensional sphere, for all correlation parameters at most 1/10 in absolute value. This inequality was conjectured (for all correlation son and Wright. Such an inequality is needed to prove sharp computational hardness of the product state Quantum MAX-CUT problem, assuming the unique games conjecture. In fact, assuming the uniqu...
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作者：Benard, Timothee; Breuillard, Emmanuel
作者单位：University of Oxford
摘要：We formulate and establish the central limit theorem for products of i.i.d. random variables on arbitrary simply connected nilpotent Lie groups, allowing a possible bias. We find that some interesting new phenomena arise in the presence of a bias: the walk spreads out at a higher rate in the ambient group, while the limiting hypoelliptic diffusion process may not always have full support. We use elementary Fourier analysis to establish our results, which include Berry-Esseen bounds under optim...
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作者：Crawford, Nicholas; Glazman, Alexander; Harel, Matan; Peled, Ron
作者单位：Technion Israel Institute of Technology; University of Innsbruck; Northeastern University; Tel Aviv University
摘要：The loop O(n) model is a family of probability measures on collections of nonintersecting loops on the hexagonal lattice, parameterized by a loop-weight n and an edge-weight x. Nienhuis predicts that, for 0 < n <2, the model exhibits two regimes separated by xc (n) = 1/2 + root 2 - n: when x < xc(n), the loop lengths have exponential tails, while when x >= xc(n), the loops are macroscopic. In this paper, we prove three results regarding the existence of long loops in the loop O(n) model: - In ...
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作者：De Vecchi, Francesco C.; Fresta, Luca; Gubinelli, Massimiliano
作者单位：University of Pavia; University of Bonn; University of Bonn; University of Oxford
摘要：Building on previous work on the stochastic analysis for Grassmann random variables, we introduce a forward-backward stochastic differential equation (FBSDE), which provides a stochastic quantisation of Grassmann measures. Our method is inspired by the so-called continuous renormalisation group, but avoids the technical difficulties encountered in the direct study of the flow equation for the effective potentials. As an application, we construct a family of weakly coupled subcritical Euclidean...
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作者：Bobkov, Sergey G.; Goetze, Friedrich
作者单位：University of Minnesota System; University of Minnesota Twin Cities; University of Bielefeld
摘要：For normalized sumsZnof i.i.d. random variables, we explore necessaryand sufficient conditions, which guarantee the normal approximation withrespect to the R & eacute;nyi divergence of infinite order. In terms of densitiespnofZn, this is a strengthened variant of the local limit theorem taking the formsupx(pn(x)-phi(x))/phi(x)-> 0asn ->infinity
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作者：Jorritsma, Joost; Komjathy, Julia; Mitsche, Dieter
作者单位：University of Oxford; Delft University of Technology; Pontificia Universidad Catolica de Chile; Centre National de la Recherche Scientifique (CNRS); Ecole Centrale de Lyon; Institut National des Sciences Appliquees de Lyon - INSA Lyon; Universite Claude Bernard Lyon 1; Universite Jean Monnet
摘要：We consider a large class of spatially-embedded random graphs that includes among others long-range percolation, continuum scale-free percolation and the age-dependent random connection model. We assume that the model is supercritical: there is an infinite component. We identify the stretch-exponent zeta is an element of (0, 1) of the decay of the cluster-size distribution. That is, with |C(0)| denoting the number of vertices in the component of the vertex at 0 is an element of R-d, we prove P...
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作者：Mossel, Elchanan; Sly, Allan; Sohn, Youngtak
作者单位：Massachusetts Institute of Technology (MIT); Princeton University
摘要：In this paper we continue to rigorously establish the predictions in ground breaking work in statistical physics by Decelle, Krzakala, Moore, Zdeborov & aacute; (Phys. Rev. E 84 (2011) 066106) regarding the block model, in particular, in the case of q = 3 and q =4 communities. We prove that, for q = 3 and q = 4, there is no computational-statistical gap if the average degree is above some constant by showing it is information theoretically impossible to detect below the Kesten-Stigum bound. Th...
