Asymptotics of k-nearest neighbor Riesz energies

2022/01/03 by Douglas P. Hardin, Edward B. Saff, Hardin, Douglas P. +3
Mathematics · Physics and Astronomy · #28A78. Secondary: 52A40 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Functional Analysis (math.FA) #Markov Chains and Monte Carlo Methods #Mathematical Approximation and Integration #Nuclear physics research studies #Primary: 31C20

paper · pdf · doi:10.48550/arxiv.2201.00474

openalex publication_date 2022/01/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We obtain new asymptotic results about systems of N particles governed by Riesz interactions involving k -nearest neighbors of each particle as N→∞. These results include a generalization to weighted Riesz potentials with external field. Such interactions offer an appealing alternative to other approaches for reducing the computational complexity of an N -body interaction. We find the first-order term of the large N asymptotics and characterize the limiting distribution of the minimizers. We also obtain results about the Γ-convergence of such interactions, and describe minimizers on the 1-dimensional flat torus in the absence of external field, for all N .

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