Trickle-Down in Localization Schemes and Applications

TL;DR

This work develops a generalized trickle-down framework within linear-tilt localization schemes to control the covariance structure of complex distributions under iterative tilting. By formulating a backward covariance recursion, the authors extend trickle-down ideas beyond high-dimensional expanders to broad probabilistic models, enabling improved mixing and sampling guarantees. The main contributions include refined spectral conditions for rapid Glauber mixing in Ising models, new Langevin and discrete-time sampling bounds for the O(N) model, semi-log-concave base-measures, and polarization-based sampling strategies on expanders, yielding near-linear-time algorithms and FPRAS results for partition functions in several settings. Together, these results advance efficient sampling for spin systems (Ising, SK, Hopfield) and continuous-spin models on expanders, with broad theoretical and practical implications for statistical physics, probabilistic combinatorics, and algorithm design.

Abstract

Trickle-down is a phenomenon in high-dimensional expanders with many important applications -- for example, it is a key ingredient in various constructions of high-dimensional expanders or the proof of rapid mixing for the basis exchange walk on matroids and in the analysis of log-concave polynomials. We formulate a generalized trickle-down equation in the abstract context of linear-tilt localization schemes. Building on this generalization, we improve the best-known results for several Markov chain mixing or sampling problems -- for example, we improve the threshold up to which Glauber dynamics is known to mix rapidly in the Sherrington-Kirkpatrick spin glass model. Other applications of our framework include improved mixing results for the Langevin dynamics in the $O(N)$ model, and near-linear time sampling algorithms for the antiferromagnetic and fixed-magnetization Ising models on expanders. For the latter application, we use a new dynamics inspired by polarization, a technique from the theory of stable polynomials.

Figures (2)

• Figure 1: $\log q_{\eta}(z)$from\ref{['thm:ising-main']}plottedusingnumericalintegrationinMathematicafor$\eta=0,0.1,\ldots,0.9,1$.Thecurvefor$\eta=0$was,beforethiswork,thebestknownboundforallvaluesof$\eta$intheIsingmodel;itasymptotesto$\infty$at$z=1$,whichistightduetothephasetransitionintheCurie-Weissmodelellis2006entropy.For$\eta=0.5$thefunction$\log q_{\eta}$asymptotesto$\infty$at$z$slightlyabove$1.18$andfor$\eta=1.0$itasymptotesaround$1.40$.
• Figure 2: $\log Q_{\eta}(z)$from\ref{['thm:ON-main']}plottedusingnumericalintegrationinMathematicafor$\eta=0,0.1,\ldots,0.9,1$.Forthe$O(N)$model,theactualfunctionboundingthecovarianceis$q_{\eta,1/N}(z)=(1/N)Q_{\eta}(z/N)$.Soforexample,inthecaseoftheXYmodelwhichis$N=2$,the$\eta=0.5$curveasymptotesto$\infty$slightlyabove$2.34$andfortheHeisenbergmodelitasymptotesto$\infty$slightlyabove$3.52$.Similarlyfor$\eta=1$,when$N=2$theresultingcurveasymptotesto$\infty$slightlyabove$2.73$forXYandataround$4.09$forHeisenberg.

Theorems & Definitions (21)

• theorem 1: Informal
• theorem 2: Informal
• theorem 3
• remark 1: Spectralinterpretation
• theorem 4
• remark 2
• theorem 5
• theorem 6: \ref{['thm:semilogconcave']}below
• theorem 7
• corollary 1