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Award ID contains: 1954337

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  1. ; (, Communications on Pure and Applied Mathematics)
    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. 
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    Free, publicly-accessible full text available June 1, 2026
  2. ; ; (, Communications in Mathematical Physics)
    Abstract Continuing our earlier work in Nam et al. (One-step replica symmetry breaking of random regular NAE-SAT I,arXiv:2011.14270, 2020), we study the random regulark-nae-satmodel in the condensation regime. In Nam et al. (2020), the (1rsb) properties of the model were established with positive probability. In this paper, we improve the result to probability arbitrarily close to one. To do so, we introduce a new framework which is the synthesis of two approaches: the small subgraph conditioning and a variance decomposition technique using Doob martingales and discrete Fourier analysis. The main challenge is a delicate integration of the two methods to overcome the difficulty arising from applying the moment method to an unbounded state space. 
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  3. ; (, Archive for rational mechanics and analysis)
    Free, publicly-accessible full text available August 13, 2026
  4. ; (, The Annals of Probability)
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  5. ; (, The Annals of Applied Probability)
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  6. ; ; (, Duke Mathematical Journal)
    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. 
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  7. ; ; (, Annales de l'Institut Henri Poincaré, Probabilités et Statistiques)
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  8. ; (, Communications in Mathematical Physics)
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  9. ; ; (, 2021 IEEE 62nd Annual Symposium on Foundation of Computer Science (FOCS))
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  10. ; (, Electronic Journal of Probability)
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