The covariance matrix of the potts model: A random cluster analysis

1994/11/14 by Christian Borgs, C. Borgs, J. T. Chayes +3
Mathematics · Physics and Astronomy · #Adaptation and Self-Organizing Systems (nlin.AO) #Condensed Matter (cond-mat) #FOS: Physical sciences #High Energy Physics - Lattice (hep-lat) #Opinion Dynamics and Social Influence #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #adap-org #cond-mat #hep-lat #nlin.AO

paper · pdf · doi:10.48550/arxiv.adap-org/9411001

52 pages, amstex, no figures

arxiv created 1994/11/14 · openalex publication_date 1994/11/14 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider the covariance matrix G^mn(x-y) of the d-dimensional q-states Potts model, rewriting it in terms of the connectivity, the finite-cluster connectivity and the infinite-cluster covariance in the random cluster repre- sentation of Fortuin and Kasteleyn. In any of the q ordered phases, we show that the matrix G^mn(x-y) has one tivial eigenvalue 0, one simple eigen- value G_\wir⁽¹⁾(x-y) and one (q-2)-fold degenerate eigenvalue G_\wir⁽²⁾(x-y). Furthermore, we identify the eigenvalues both in terms of representations of the unbroken symmetry group of the model, and in terms of connectivities and cluster covariances, thereby attributing algebraic signifi- cance to these stochastic geometric quantities. In addition to establishing the existence of the correlation lengths ξ_\wir⁽¹⁾ and ξ_\wir⁽²⁾ corresponding to G_\wir⁽¹⁾(x-y) and G_\wir⁽²⁾(x-y), we show that ξ_\wir⁽¹⁾(β)≥ ξ_\wir⁽²⁾(β) for all inverse tempera- tures β. For dimension d=2 and q ≥ 1, we establish a duality relation between ξ_\wir⁽²⁾ and ξ_\free, the correlation length of the two-point function with free boundary conditions: We show ξ_\wir⁽²⁾(β) = (1)/(2) ξ_\free(β^∗) for all β≥ β_o, where β^∗ is the dual inverse temperature and β_o is the self-dual point. In order to prove the above results, we introduce two new inequalities. The first is similar to the FKG inequality, but holds for events which are neither increasing nor decreasing, and replaces independence in the standard percolation model; the second replaces the van den Berg - Kesten inequality.

Related

• Discontinuity of the phase transition for the planar random-cluster and Potts models with q>4
• The self-dual point of the two-dimensional random-cluster model is critical for q≥ 1
• Study of Cluster Fluctuations in Two-dimensional q-State Potts Model
• Backbone three-point correlation function in the two-dimensional Potts model
• The Random-Cluster Model