arXiv subjects

Nikolai Kuchumov

Publications and source records attributed to Nikolai Kuchumov.

3 recordsLinked to original sources

arXiv2026

Limit shapes and harmonic tricks

This article has two main goals. First, it provides a self-contained exposition of the tangent plane method for the dimer model - a technique for analyzing arctic curves and limit shapes introduced by R. Kenyon and I. Prause (2020). Second, it extends this method to multiply connected domains through a nontrivial computation of the frozen boundary for the Aztec diamond with a hole. This computation yields the first explicit parametrization in terms of elliptic functions of a family of arctic curves of a multiply-connected region indexed by the height change (hole height). We also derive and visualize the corresponding limit height functions.

math-ph↗

arXiv2023

A variational principle for domino tilings of multiply-connected domains

We study random domino tilings of a multiply-connected domain with a height function defined on the universal covering space of the domain. We prove a large deviation principle for the height function in two asymptotic regimes. The first regime covers all domino tilings of the domain. We also prove a law of large numbers for height change in this regime. The second regime covers domino tilings with a given asymptotic height change $r$.

math.PR↗

arXiv2017

Limit shapes for the dimer model

We prove the existence of a limit shape for the dimer model on planar periodic bipartite graphs with an arbitrary fundamental domain and arbitrary periodic weights. This proof is based on a variational principle that uses the locality of the model and the compactness of the space of states.

math-ph↗