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Coexistence, enhancements and short loops in random walk loop soups. (English) Zbl 1543.60114

Summary: We consider a general random walk loop soup which includes, or is related to, several models of interest, such as the spin \(\mathrm{O}(N)\) model, the double dimer model and the Bose gas. The analysis of this model is challenging because of the presence of spatial interactions between the loops. For this model it is known from earlier work [A. Quitmann and L. Taggi, Commun. Math. Phys. 400, No. 3, 2081–2136 (2023; Zbl 07692743)] that macroscopic loops occur in dimension three and higher when the inverse temperature is large enough. Our first result is that, on the \(d\)-dimensional lattice, the presence of repulsive interactions is responsible for a shift of the critical inverse temperature, which is strictly greater than \(\frac{1}{2d}\), the critical value in the noninteracting case. Our second result is that a positive density of microscopic loops exists for all values of the inverse temperature. This implies that, in the regime in which macroscopic loops are present, microscopic and macroscopic loops coexist. We show that, even though the increase of the inverse temperature leads to an increase of the total loop length, the density of microscopic loops is uniformly bounded from above in the inverse temperature. Our last result is confined to the special case in which the random walk loop soup is the one associated to the spin \(\mathrm{O}(N)\) model with arbitrary integer values of \(N \geq 2\) and states that, on \(\mathbb{Z}^2\), the probability that two vertices are connected by a loop decays at least polynomially fast with their distance.

MSC:

60K35 Interacting random processes; statistical mechanics type models; percolation theory
82C41 Dynamics of random walks, random surfaces, lattice animals, etc. in time-dependent statistical mechanics

Citations:

Zbl 07692743

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