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Abstract We study a stochastic Laplacian growth model, where a set grows according to a reflecting Brownian motion in stopped at level sets of its boundary local time. We derive a scaling limit for the leading‐order behavior of the growing boundary (i.e., “interface”). It is given by a geometric flow‐typepde. It is obtained by an averaging principle for the reflecting Brownian motion. We also show that this geometric flow‐typepdeis locally well‐posed, and its blow‐up times correspond to changes in the diffeomorphism class of the growth model. Our results extend those of Dembo et al., which restricts to star‐shaped growth domains and radially outwards growth, so that in polar coordinates, the geometric flow transforms into a simpleodewith infinite lifetime. Also, we remove the “separation of scales” assumption that was taken in Dembo et al.; this forces us to understand the local geometry of the growing interface. 

We develop a quantitative large deviations theory for random hypergraphs, which rests on tensor decomposition and counting lemmas under a novel family of cut-type norms. As our main application, we obtain sharp asymptotics for joint upper and lower tails of homomorphism counts in the r-uniform Erdo ̋s–Rényi hypergraph for any fixed r≥2, generalizing and improving on previous results for the Erdo ̋s–Rényi graph (r=2). The theory is sufficiently quantitative to allow the density of the hypergraph to vanish at a polynomial rate, and additionally yields tail asymptotics for other nonlinear functionals, such as induced homomorphism counts. 

Abstract Consider $$(X_{i}(t))$$ ( X i ( t ) ) solving a system of N stochastic differential equations interacting through a random matrix $${\mathbf {J}} = (J_{ij})$$ J = ( J ij ) with independent (not necessarily identically distributed) random coefficients. We show that the trajectories of averaged observables of $$(X_i(t))$$ ( X i ( t ) ) , initialized from some $$\mu $$ μ independent of  $${\mathbf {J}}$$ J , are universal, i.e., only depend on the choice of the distribution $$\mathbf {J}$$ J through its first and second moments (assuming e.g., sub-exponential tails). We take a general combinatorial approach to proving universality for dynamical systems with random coefficients, combining a stochastic Taylor expansion with a moment matching-type argument. Concrete settings for which our results imply universality include aging in the spherical SK spin glass, and Langevin dynamics and gradient flows for symmetric and asymmetric Hopfield networks.