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1. Critical value asymptotics for the contact process on random graphs

   https://doi.org/10.1090/tran/8399  

   Nam, Danny; Nguyen, Oanh; Sly, Allan (January 2022, Transactions of the American Mathematical Society)

   Recent progress in the study of the contact process (see Shankar Bhamidi, Danny Nam, Oanh Nguyen, and Allan Sly [Ann. Probab. 49 (2021), pp. 244–286]) has verified that the extinction-survival threshold λ 1 \lambda _1 on a Galton-Watson tree is strictly positive if and only if the offspring distribution ξ \xi has an exponential tail. In this paper, we derive the first-order asymptotics of λ 1 \lambda _1 for the contact process on Galton-Watson trees and its corresponding analog for random graphs. In particular, if ξ \xi is appropriately concentrated around its mean, we demonstrate that λ 1 ( ξ ) ∼ 1 / E ξ \lambda _1(\xi ) \sim 1/\mathbb {E} \xi as E ξ → ∞ \mathbb {E}\xi \rightarrow \infty , which matches with the known asymptotics on d d -regular trees. The same results for the short-long survival threshold on the Erdős-Rényi and other random graphs are shown as well. 

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2. Riesz means asymptotics for Dirichlet and Neumann Laplacians on Lipschitz domains

   https://doi.org/10.1007/s00222-025-01352-x  

   Frank, Rupert L; Larson, Simon (September 2025, Inventiones mathematicae)

   Abstract We consider the eigenvalues of the Dirichlet and Neumann Laplacians on a bounded domain with Lipschitz boundary and prove two-term asymptotics for their Riesz means of arbitrary positive order. Moreover, when the underlying domain is convex, we obtain universal, non-asymptotic bounds that correctly reproduce the two leading terms in the asymptotics and depend on the domain only through simple geometric characteristics. Important ingredients in our proof are non-asymptotic versions of various Tauberian theorems. 

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3. Counting special Lagrangian classes and semistable mukai vectors for K3 surfaces

   https://doi.org/10.1007/s10711-023-00823-w  

   Athreya, Jayadev S; Fan, Yu-Wei; Lee, Heather (October 2023, Geometriae Dedicata)

   Motivated by the study of the growth rate of the number of geodesics in flat surfaces with bounded lengths, we study generalizations of such problems for K3 surfaces. In one gener- alization, we give a result regarding the upper bound on the asymptotics of the number of classes of irreducible special Lagrangians in K3 surfaces with bounded period integrals. In another generalization, we give the exact leading term in the asymptotics of the number of Mukai vectors of semistable coherent sheaves on algebraic K3 surfaces with bounded central charges, with respect to generic Bridgeland stability conditions. 

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4. A stochastic spatial model for the sterile insect control strategy

   https://doi.org/10.1016/j.spa.2022.11.018  

   Huang, Xiangying; Durrett, Rick (March 2023, Stochastic Processes and their Applications)

   In the system we study, 1's and 0's represent occupied and vacant sites in the contact process with births at rate $$\lambda$$ and deaths at rate 1. $-1$'s are sterile individuals that do not reproduce but appear spontaneously on vacant sites at rate $$\alpha$$ and die at rate $$\theta\alpha$$. We show that the system (which is attractive but has no dual) dies out at the critical value and has a nontrivial stationary distribution when it is supercritical. Our most interesting results concern the asymptotics when $$\alpha\to 0$$. In this regime the process resembles the contact process in a random environment. 

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5. Phylogeography and paleoclimatic range dynamics explain variable outcomes to contact across a species' range

   https://doi.org/10.1111/mec.17450  

   Lamb, Keric; Debban, Catherine L; Galloway, Laura F (August 2024, Molecular Ecology)

   Abstract Replicability of divergence after contact is a poorly characterized process, particularly in the contexts of phylogeography and postglacial range dynamics within species. Using contact zones located at the leading‐, mid‐ and rear‐edges of a species' range, we examined variation in outcomes to contact between divergent lineages ofCampanula americana. We investigated whether contact zones vary in quantity and directionality of gene flow, how phylogeographic structure differs between contact zones, and how historic range dynamics may affect outcomes to contact. We found that all contact zones formed at similar times via primary contact yet detected significant admixture in only the rear‐edge (RE) contact zone. In the northern leading‐edge contact zone and the mid‐range Virginia contact zone, gene flow was minimal and asymmetric. In the southern RE contact zone, gene flow was strong and symmetric. Asymmetric admixture in the leading‐edge and Virginia contact zones matches the directionality of a known cosmopolitan cytonuclear incompatibility between lineages ofC. americana. Our results emphasize the dependence of speciation processes on phylogeographic structure, evolutionary history and range dynamics. 

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