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作者：Hairer, Martin; Le, Khoa; Rosati, Tommaso
作者单位：Imperial College London; Swiss Federal Institutes of Technology Domain; Ecole Polytechnique Federale de Lausanne; Technical University of Berlin; University of Leeds; University of Warwick
摘要：We consider the Allen-Cahn equation ?(t)u - ?u = u - u(3) with a rapidly mixing Gaussian field as initial condition. We show that provided that the amplitude of the initial condition is not too large, the equation generates fronts described by nodal sets of the Bargmann-Fock Gaussian field, which then evolve according to mean curvature flow.
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作者：Penrose, Mathew D.
作者单位：University of Bath
摘要：Let X-1, X-2, . . . be independent random uniform points in a bounded domain A subset of R-d with smooth boundary. Define the coverage threshold Rn to be the smallest r such that A is covered by the balls of radius r centred on X-1, . . . , X-n. We obtain the limiting distribution of R(n )and also a strong law of large numbers for R-n in the large -n limit. For example, if A has volume 1 and perimeter |& part; A|, if d = 3 then P[n pi R-n(3) - log n - 2 log(log n) <= x] converges to exp(-2(-4)...
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作者：Yu, Lei
作者单位：Nankai University; Nankai University
摘要：Given a convex function 0 : [0, 1] ? R and the mean ? f (X) = a ? [0, 1], which Boolean function f maximizes the 0-stability E[F (T-? f(X))] of f ? Here X is a random vector uniformly distributed on the discrete cube {-1, 1}n and T(? )is the Bonami-Beckner operator. Special cases of this problem include the (symmetric and asymmetric) a-stability problems and the Most Informative Boolean Function problem. In this paper, we provide several upper bounds for the maximal 0-stability. When specializ...
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作者：Ahn, Andrew
作者单位：Cornell University
摘要：We establish universality for the largest singular values of products of random matrices with right unitarily invariant distributions, in a regime where the number of matrix factors and size of thematrices tend to infinity simultaneously. The behavior of the largest log singular values coincides with the large N limit of Dyson Brownian motion with a characteristic drift vector consisting of equally spaced coordinates, which matches the large N limit of the largest log singular values of Browni...
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作者：Subag, Eliran
作者单位：Weizmann Institute of Science
摘要：We compute the free energy at all temperatures for the spherical pure p-spin models from the generalized Thouless-Anderson-Palmer representation. This is the first example of a mixed p-spin model for which the free energy is computed in the whole replica symmetry breaking phase, without appealing to the famous Parisi formula.
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作者：Foutel-Rodier, Felix; Schertzer, Emmanuel
作者单位：University of Oxford; University of Vienna
摘要：Consider a branching Markov process with values in some general type space. Conditional on survival up to generation N, the genealogy of the extant population defines a random marked metric measure space, where individuals are marked by their type and pairwise distances are measured by the time to the most recent common ancestor. In the present manuscript, we devise a general method of moments to prove convergence of such genealogies in the Gromov-weak topology when N ? 8. Informally, the mome...
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作者：Rosenzweig, Matthew; Staffilani, Gigliola
作者单位：Massachusetts Institute of Technology (MIT)
摘要：Aggregation equations, such as the parabolic-elliptic Patlak-Keller-Segel model, are known to have an optimal threshold for global existence versus finite-time blow-up. In particular, if the diffusion is absent, then all smooth solutions with finite second moment can exist only locally in time. Nevertheless, one can ask whether global existence can be restored by adding a suitable noise to the equation, so that the dynamics are now stochastic. Inspired by the work of Buckmaster et al. (Int Mat...
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作者：Schonbauer, Philipp
作者单位：Imperial College London
摘要：We study Malliavin differentiability of solutions to sub-critical singular parabolic stochastic partial differential equations (SPDEs) and we prove the existence of densities for a class of singular SPDEs. Both of these results are implemented in the setting of regularity structures. For this we construct renormalized models in situations where some of the driving noises are replaced by deterministic Cameron-Martin functions, and we show Lipschitz continuity of these models with respect to the...
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作者：Blanc-Renaudie, Arthur
作者单位：Sorbonne Universite
摘要：We introduce a new stick-breaking construction for inhomogeneous continuum random trees. This new construction allows us to prove the necessary and sufficient condition for compactness conjectured by Aldous et al. (Probab Theory Relat Fields 129(2):182-218, 2004) by comparison with Levy trees. We also compute the fractal dimensions (Minkowski, Packing, Hausdorff).
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作者：Lacker, Daniel; Ramanan, Kavita; Wu, Ruoyu
作者单位：Columbia University; Brown University; Iowa State University
摘要：Consider a system of homogeneous interacting diffusive particles labeled by the nodes of a unimodular Galton-Watson tree, where the state of each node evolves infinitesimally like a d-dimensional diffusion whose drift coefficient depends on (the histories of) its own state and the states of neighboring nodes, and whose diffusion coefficient depends only on (the history of) its own state. Under suitable regularity assumptions on the coefficients, an autonomous characterization is obtained for t...
