{"ID":2858512,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2510.06549","arxiv_id":"2510.06549","title":"Trickle-down Theorems via C-Lorentzian Polynomials II: Pairwise Spectral Influence and Improved Dobrushin's Condition","abstract":"Let $μ$ be a probability distribution on a multi-state spin system on a set $V$ of sites; equivalently, a $d$-partite simplicial complex with distribution $μ$ on maximal faces. For any pair of vertices $u,v\\in V$, define the pairwise spectral influence $\\mathcal{I}_{u,v}$ as follows. Let $σ$ be a choice of spins $s_w\\in S_w$ for every $w\\in V\\setminus\\{u,v\\}$, and construct a matrix in $\\mathbb{R}^{(S_u\\cup S_v)\\times (S_u\\cup S_v)}$ where for any $s_u\\in S_u, s_v\\in S_v$, the $(us_u,vs_v)$-entry is the probability that $s_v$ is the spin of $v$ conditioned on $s_u$ being the spin of $u$ and on $σ$. Then $\\mathcal{I}_{u,v}$ is the maximal second eigenvalue of this matrix, over all choices of spins for all $w\\in V\\setminus\\{u,v\\}$. Equivalently, $\\mathcal{I}_{u,v}$ is the maximum local spectral expansion of links of codimension $2$ that include a spin for every $w \\in V \\setminus \\{u,v\\}$. We show that if the largest eigenvalue of the pairwise spectral influence matrix with entries $\\mathcal{I}_{u,v}$ is bounded away from 1, i.e. $λ_{\\max}(\\mathcal{I})\\leq 1-ε$ (and $X$ is connected), then the Glauber dynamics mixes rapidly and generate samples from $μ$. This improves/generalizes the classical Dobrushin's influence matrix as the $\\mathcal{I}_{u,v}$ lower-bounds the classical influence of $u\\to v$. As an application, we prove that the Glauber dynamics mixes rapidly up to (approximately) the phase transition for the multi-state hardcore model--a widely studied model in telecommunication networks and statistical physics (generalizing the hardcore model) introduced by Mazel and Suhov. As a by-product of our results, we also prove improved/almost optimal trickle-down theorems for partite simplicial complexes. Our proof builds on the trickle-down theorems via $\\mathcal{C}$-Lorentzian polynomials machinery recently developed by the authors and Lindberg.","short_abstract":"Let $μ$ be a probability distribution on a multi-state spin system on a set $V$ of sites; equivalently, a $d$-partite simplicial complex with distribution $μ$ on maximal faces. For any pair of vertices $u,v\\in V$, define the pairwise spectral influence $\\mathcal{I}_{u,v}$ as follows. Let $σ$ be a choice of spins $s_w\\i...","url_abs":"https://arxiv.org/abs/2510.06549","url_pdf":"https://arxiv.org/pdf/2510.06549v3","authors":"[\"Jonathan Leake\",\"Shayan Oveis Gharan\"]","published":"2025-10-08T01:00:48Z","proceeding":"math.CO","tasks":"[\"math.CO\",\"cs.CC\",\"cs.DS\"]","methods":"[]","has_code":false}