We consider the 4D fixed length lattice Higgs model with Wilson action for the gauge field and structure group Zn. We study Wilson line observables in the strong coupling regime and compute their asymptotic behavior with error estimates. Our analysis is based on a high-temperature representation of the lattice Higgs measure combined with Poisson approximation. We also give a short proof of the folklore result that Wilson line (and loop) expectations exhibit perimeter law decay whenever the Higgs field coupling constant is positive.

We consider the lattice Higgs model on ℤ4 in the fixed length limit, with structure group given by ℤn for n ⩾ 2. We compute the expected value of the Wilson loop observable to leading order when the inverse temperature and hopping parameter are both sufficiently large compared to the length of the loop. The leading order term is expressed in terms of a quantity arising from the related but much simpler ℤn model, which reduces to the usual Ising model when n = 2. As part of the proof, we construct a coupling between the lattice Higgs model and the ℤn model.

In Steif and Tykesson (J Prob 16:899–955, 2019), the authors introduced the so-called general divide and color models. One of the best-known examples of such a model is the Ising model with external field h= 0 , which has a color representation given by the random cluster model. In this paper, we give necessary and sufficient conditions for this color representation to be unique. We also show that if one considers the Ising model on a complete graph, then for many h> 0 , there is no color representation. This shows, in particular, that any generalization of the random cluster model which provides color representations of Ising models with external fields cannot, in general, be a generalized divide and color model. Furthermore, we show that there can be at most finitely many β> 0 at which the random cluster model can be continuously extended to a color representation for h≠ 0.

In this paper, we study lattice gauge theory on Z4 with finite Abelian structure group. When the inverse coupling strength is sufficiently large, we use ideas from disagreement percolation to give an upper bound on the decay of correlations of local functions. We then use this upper bound to compute the leading-order term for both the expected value of the spin at a given plaquette as well as for the two-point correlation function. Moreover, we give an upper bound on the dependency of the size of the box on which the model is defined. The results in this paper extend and refine results by Chatterjee and Borgs. 

We consider lattice gauge theories on Z4 with Wilson action and structure group Zn. We compute the expectation of Wilson loop observables to leading order in the weak coupling regime, extending and refining a recent result of Chatterjee. Our proofs use neither duality relations nor cluster expansion techniques.

We study the natural linear operators associated to divide and color (DC) models. The degree of nonuniqueness of the random partition yielding a DC model is directly related to the dimension of the kernel of these linear operators. We determine exactly the dimension of these kernels as well as analyze a permutation-invariant version. We also obtain properties of the solution set for certain parameter values which will be important in (1) showing that large threshold discrete Gaussian free fields are DC models and in (2) analyzing when the Ising model with a positive external field is a DC model, both in future work. However, even here, we give an application to the Ising model on a triangle.

In [10], Jonasson and Steif conjectured that no non-degenerate sequence of transitive Boolean functions (f(n))(n >= 1) with lim(n ->infinity) I(f(n)) = infinity could be tame (with respect to some (p(n))(n >= 1)). In a companion paper [5], the author showed that this conjecture in its full generality is false, by providing a counter-example for the case when, at the same time, lim(n ->infinity) np(n) = infinity and lim(n ->infinity) n(alpha)p(n) = 0 for some alpha is an element of (0, 1). In this paper we show that with slightly different assumptions, the conclusion of the conjecture holds when the sequence (p(n))(n >= 1) is bounded away from zero and one.

Using formulas for certain quantities involving stable vectors, due to I. Molchanov, and in some cases utilizing the so-called divide and color modelwe prove that certain families of integrals which, ostensibly, depend on a parameter are in fact independent of this parameter.

We develop a formula for the power-law decay of various sets for symmetric stable random vectors in terms of how many vectors from the support of the corresponding spectral measure are needed to enter the set. One sees different decay rates in “different directions”, illustrating the phenomenon of hidden regular variation. We give several examples and obtain quite varied behavior, including sets which do not have exact power-law decay. 

Given a sequence of Boolean functions (f(n))(n)>= , f(n): {0,1}(n) -> {0,1}, and a sequence (X-(n))(n >= 1) of continuous time p(n)-biased random walks X-(n) = (X-t((n)))(t >= 0) on {0,1}(n), let Cr, be the (random) number of times in (0,1) at which the process (f(n)(X-t))(t >= 0) changes its value. In [7], the authors conjectured that if (f(n))(n >= 1) is non-degenerate, transitive and satisfies lim(n ->infinity) E[C-n] = infinity, then (C-n)(n >= 1)is not tight. We give an explicit example of a sequence of Boolean functions which disproves this conjecture.