Dennis Belotserkovskiy

arXiv2026

Uniqueness of the second eigenspace of the interchange process

The spectral gap theorem of Caputo, Liggett, and Richthammer states that on any connected weighted graph, the second eigenvalue of the interchange process equals the second eigenvalue of the random walk process. We characterize the corresponding eigenspace in the regular representation of the symmetric group. Except when the graph is a $4$-cycle whose four edge weights are equal, the entire second eigenspace lies in the direct sum of copies of the standard representation and is generated by copies of the second eigenspace of the random-walk Laplacian. The proof refines the octopus induction scheme by isolating the summand of the restricted representation that can support equality, and analyzing it in explicit two-subset and exterior-square coordinates.

math.PR↗

arXiv2025

Asymptotic invariants of symbolic powers of binomial edge ideals

To a graph $G$ one associates the binomial edge ideal $J_G$ generated by a collection of binomials corresponding to the edges of $G$. In this paper, we study the asymptotic behavior of symbolic powers of $J_G$, its lexicographic initial ideal $\mathrm{in}_<(J_G)$, and its multigraded generic initial ideal $\mathrm{gin}(J_G)$. We focus on the Waldschmidt constant, $\widehatα$, and asymptotic regularity, $\widehat{\mathrm{reg}}$, which capture linear growth of minimal generator degrees and Castelnuovo--Mumford regularity. We explicitly compute $\widehatα(J_G)$ and $\widehatα(\mathrm{in}_<(J_G))$, and compare the Betti numbers of the symbolic powers of $J_G$ and $J_H$, where $H$ is a subgraph of $G$. To analyze $\mathrm{in}_<(J_G)$ and $\mathrm{gin}(J_G)$, we use the symbolic polyhedron, a convex polyhedron that encodes the elements of the symbolic powers of a monomial ideal. We determine its vertices via $G$'s induced connected subgraphs and show that $\widehatα(\mathrm{gin}(J_G))=\widehatα(I_G)$, where $I_G$ is the edge ideal of $G$. This yields an alternate proof of known bounds for $\widehatα(I_G)$ in terms of $G$'s clique number and chromatic number.

math.AC↗