We revisit the cluster expansion for Ising lattice gauge theory on Z^m, m ≥ 3, with Wilson’s action, at a fixed inverse temperature β in the low-temperature regime. We prove existence and analyticity of the infinite volume limit of the free energy and compute the first few terms in its expansion in powers of e^β. We further analyze Wilson loop expectations and derive an estimate that shows how the lattice scale geometry of a loop is reflected in the large β asymptotic expansion. Specializing to axis parallel rectangular loops γT,R with side-lengths T and R, we consider the limiting function Vβ(R) := limT→∞ − _T¹ log 〈Wγ_T,R>_β, known as the static quark potential in the physics literature. We verify the existence of the limit (with an estimate on the convergence rate) and compute the first few terms in the expansion in powers of e^−β. As a consequence, a strong version of the perimeter law follows. We also treat − log 〈Wγ_T,R〉β/(T + R) as T, R tend to infinity simultaneously and give analogous estimates.

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 study percolative properties of excursion processes and the discrete Gaussian free field (dGFF) in the planar unit disk. We consider discrete excursion clouds, defined using random walks as a two-dimensional version of random interlacements, as well as its scaling limit, defined using Brownian motion. We prove that the critical parameters associated to vacant set percolation for the two models are the same and equal to pi /3. The value is obtained from a Schramm-Loewner evolution (SLE) computation. Via an isomorphism theorem, we use a generalization of the discrete result that also involves a loop soup (and an SLE computation) to show that the critical parameter associated to level set percolation for the dGFF is strictly positive and smaller than root pi/2. In particular this entails a strict inequality of the type h(*) < root 2u(*) between the critical percolation parameters of the dGFF and the two-dimensional excursion cloud. Similar strict inequalities are conjectured to hold in a general transient setup.

We study foliations by chord–arc Jordan curves of the twice punctured Riemann sphere (Formula presented.) using the Loewner–Kufarev equation. We associate to such a foliation a function on the plane that describes the “local winding” along each leaf. Our main theorem is that this function has finite Dirichlet energy if and only if the Loewner driving measure (Formula presented.) has finite Loewner–Kufarev energy, defined by (Formula presented.) whenever (Formula presented.) is of the form (Formula presented.), and set to (Formula presented.) otherwise. Moreover, if either of these two energies is finite, they are equal up to a constant factor, and in this case, the foliation leaves are Weil–Petersson quasicircles. This duality between energies has several consequences. The first is that the Loewner–Kufarev energy is reversible, that is, invariant under inversion and time reversal of the foliation. Furthermore, the Loewner energy of a Jordan curve can be expressed using the minimal Loewner–Kufarev energy of those measures that generate the curve as a leaf. This provides a new and quantitative characterization of Weil–Petersson quasicircles. Finally, we consider conformal distortion of the foliation and show that the Loewner–Kufarev energy satisfies an exact transformation law involving the Schwarzian derivative. The proof of our main theorem uses an isometry between the Dirichlet energy space on the unit disc and (Formula presented.) that we construct using Hadamard's variational formula expressed by means of the Loewner–Kufarev equation. Our results are related to (Formula presented.) -parameter duality and large deviations of Schramm–Loewner evolutions coupled with Gaussian random fields.

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.

Critical statistical mechanics and Conformal Field Theory (CFT) are conjecturally connected since the seminal work of Beliavin et al. (Nucl Phys B 241(2):333–380, 1984). Both exhibit exactly solvable structures in two dimensions. A long-standing question (Itoyama and Thacker in Phys Rev Lett 58:1395–1398, 1987) concerns whether there is a direct link between these structures, that is, whether the Virasoro algebra representations of CFT, the distinctive feature of CFT in two dimensions, can be found within lattice models of statistical mechanics. We give a positive answer to this question for the discrete Gaussian free field and for the Ising model, by connecting the structures of discrete complex analysis in the lattice models with the Virasoro symmetry that is expected to describe their scaling limits. This allows for a tight connection of a number of objects from the lattice model world and the field theory one. In particular, our results link the CFT local fields with lattice local fields introduced in Gheissari et al. (Commun Math Phys 367(3):771–833, 2019) and the probabilistic formulation of the lattice model with the continuum correlation functions. Our construction is a decisive step towards establishing the conjectured correspondence between the correlation functions of the CFT fields and those of the lattice local fields. In particular, together with the upcoming (Chelkak et al. in preparation), our construction will complete the picture initiated in Hongler and Smirnov (Acta Math 211:191–225, 2013), Hongler (Conformal invariance of ising model correlations, 2012) and Chelkak et al. (Annals Math 181(3):1087–1138, 2015), where a number of conjectures relating specific Ising lattice fields and CFT correlations were proven. 

Two-valued sets are local sets of the 2D Gaussian free field (GFF) that can be thought of as representing all points of the domain that may be connected to the boundary by a curve on which the GFF takes values only in [-a, b]. Two-valued sets exist whenever a+b >= 2 lambda, where lambda depends explicitly on the normalization of the GFF. We prove that the almost sure Hausdorff dimension of the two-valued set A(-a,b) equals d = 2- 2 lambda(2)/(a+ b)(2). For the key two-point estimate needed to give the lower bound on dimension, we use the real part of a "vertex field" built from the purely imaginary Gaussian multiplicative chaos. We also construct a non-trivial d-dimensional measure supported on A(-a,b) and discuss its relation with the d-dimensional conformal Minkowski content of A(-a,b).

We revisit the convergence of loop-erased random walk, LERW, to SLE2 when the curves are parametrized by capacity. We construct a Markovian coupling of the driving processes and Loewner chains for the chordal version of LERW and chordal SLE2 based on the Green's function for LERW as martingale observable and using an elementary discrete-time Loewner "difference" equation. We keep track of error terms and obtain power-law decay. This coupling is different than the ones previously considered in this context, e.g., in that each of the processes has the domain Markov property at mesoscopic capacity time increments, given the sigma algebra of the coupling. At the end of the paper we discuss in some detail a version of Skorokhod embedding. Our recent work on the convergence of LERW parametrized by length to SLE2 parameterized by Minkowski content uses specific features of the coupling constructed here.

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.

Loop-erased random walk (LERW) is one of the most well-studied critical lattice models. It is the self-avoiding random walk that one gets after erasing the loops from a simple random walk in order or, alternatively, by considering the branches in a spanning tree chosen uniformly at random. We prove that planar LERW parameterized by renormalized length converges in the lattice-size scaling limit to SLE2 parameterized by 5/4-dimensional Minkowski content. In doing this, we also provide a method for proving similar convergence results for other models converging to SLE. Besides the main theorem, several of our results about LERW are of independent interest. We give, for example, two-point estimates, estimates on maximal content, and a "separation lemma" for two-sided LERW.