Exponential decay of loop lengths in the loop O(n) model with large\n n

2014/12/29 by Hugo Duminil‐Copin, Ron Peled, Duminil-Copin, Hugo +5 · 2 citations
Mathematics · Physics and Astronomy · #60K35 #82B05 #82B20 #82B26 #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR) #Random Matrices and Applications #Stochastic processes and statistical mechanics #Theoretical and Computational Physics

paper · pdf · doi:10.48550/arxiv.1412.8326

openalex publication_date 2014/12/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The loop O(n) model is a model for a random collection of non-intersecting\nloops on the hexagonal lattice, which is believed to be in the same\nuniversality class as the spin O(n) model. It has been conjectured that both\nthe spin and the loop O(n) models exhibit exponential decay of correlations\nwhen n>2. We verify this for the loop O(n) model with large parameter n,\nshowing that long loops are exponentially unlikely to occur, uniformly in the\nedge weight x. Our proof provides further detail on the structure of typical\nconfigurations in this regime. Putting appropriate boundary conditions, when\nnx⁶ is sufficiently small, the model is in a dilute, disordered phase in\nwhich each vertex is unlikely to be surrounded by any loops, whereas when\nnx⁶ is sufficiently large, the model is in a dense, ordered phase which is a\nsmall perturbation of one of the three ground states.\n

Citations

• An introduction to the dimer model

Cited by

• Indistinguishability for recurrent clusters
• Dimerization in O(n)-invariant quantum spin chains

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