Site Monotonicity and Uniform Positivity for Interacting Random Walks and the Spin O(N) Model with Arbitrary N

Abstract

We provide a uniformly-positive point-wise lower bound for the two-point function of the classical spin O(N ) model on the torus of Z d , d ≥ 3, when N ∈ N >0 and the inverse temperature β is large enough. This is a new result when N > 2 and extends the classical result of Fröhlich, Simon and Spencer (1976). Our bound follows from a new site-monotonicity property of the two-point function which is of independent interest and holds not only for the spin O(N ) model with arbitrary N ∈ N >0 , but for a wide class of systems of interacting random walks and loops, including the loop O(N ) model, random lattice permutations, the dimer model, the double-dimer model, and the loop representation of the classical spin O(N ) model. *

Exponential decay of transverse correlations for O(N) spin systems and related models

“…The proof is very similar to [13,Proposition 2.3], where only the case of A consisting of two points and no external magnetic field was considered, and to [3,Proposition 6.1], where other type of correlations and no external magnetic field were considered. To begin, for A ⊂ V we define (ϕ) .…”

“…, N } and the measure involves an on-site weight function that penalises large numbers of overlaps. This representation corresponds to a combination of the ones introduced in [3,13], which are in turn related to the one of Brydges, Fröhlich and Spencer [7], and the random current representation of the Ising model [1]. In our representation a ghost vertex, denoted by g, is added to the graph, with edges to each other vertex representing the external field.…”

“…We refer to this representation as the Random Path Model (RPM). This representation corresponds to a combination of the one introduced in [3,13], which was also used in [18] in the study of the dimer model in Z d , d ≥ 3, and of the random current representation of the Ising model [1]. Two key aspects of the representation are pairings and colours, these are two ingredients which are not present (or necessary) in the N = 1 case [1], the well-known Ising model, but which play a crucial role in our analysis, which involves the N > 1 cases.…”

Site-Monotonicity Properties for Reflection Positive Measures with Applications to Quantum Spin Systems

“…B Lorenzo Taggi lorenzo.taggi@wias-berlin.de; lorenzo.taggi@uniroma1.it Benjamin Lees benjamin.lees@bristol.ac.uk This generalises the monotonicity and positivity results of [13] to a very general system. This general result has the following implications.…”

Loop Correlations in Random Wire Models

“…They can be written as ratios of loop partition functions, with the numerator involving "open" configurations of links where 2n x and 2n y are odd. See [1,20] for the Ising random currents and [15,28] for the related loop model for O(N ) spin systems. But these correlations do not have a direct probability meaning and we ignore them in this article.…”