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Register a new userConformal invariance of double random currents and the XOR-Ising model II: tightness and properties in the discrete
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This is the second of two papers devoted to the proof of conformal invariance of the critical double random current and the XOR-Ising model on the square lattice. More precisely, we show the convergence of loop ensembles obtained by taking the cluster boundaries in the sum of two independent currents both with free or wired boundary conditions, and in the XOR-Ising models with free and plus/plus boundary conditions. Therefore we establish Wilsons conjecture on the XOR-Ising model. The strategy, which to the best of our knowledge is different from previous proofs of conformal invariance, is based on the characterization of the scaling limit of these loop ensembles as certain local sets of the continuum Gaussian Free Field. In this paper, we derive crossing properties of the discrete models required to prove this characterization.
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This is the first of two papers devoted to the proof of conformal invariance of the critical double random current and the XOR-Ising models on the square lattice. More precisely, we show the convergence of loop ensembles obtained by taking the cluster boundaries in the sum of two independent currents with free and wired boundary conditions, and in the XOR-Ising models with free and plus/plus boundary conditions. Therefore we establish Wilsons conjecture on the XOR-Ising model. The strategy, which to the best of our knowledge is different from previous proofs of conformal invariance, is based on the characterization of the scaling limit of these loop ensembles as certain local sets of the Gaussian Free Field. In this paper, we identify uniquely the possible subsequential limits of the loop ensembles. Combined with the second paper, this completes the proof of conformal invariance.
Given a sequence of lattice approximations $D_Nsubsetmathbb Z^2$ of a bounded continuum domain $Dsubsetmathbb R^2$ with the vertices outside $D_N$ fused together into one boundary vertex $varrho$, we consider discrete-time simple random walks in $D_Ncup{varrho}$ run for a time proportional to the expected cover time and describe the scaling limit of the exceptional level sets of the thick, thin, light and avoided points. We show that these are distributed, up a spatially-dependent log-normal factor, as the zero-average Liouville Quantum Gravity measures in $D$. The limit law of the local time configuration at, and nearby, the exceptional points is determined as well. The results extend earlier work by the first two authors who analyzed the continuous-time problem in the parametrization by the local time at $varrho$. A novel uniqueness result concerning divisible random measures and, in particular, Gaussian Multiplicative Chaos, is derived as part of the proofs.
This is the second part of a three part series abut delocalization for band matrices. In this paper, we consider a general class of $Ntimes N$ random band matrices $H=(H_{ij})$ whose entries are centered random variables, independent up to a symmetry constraint. We assume that the variances $mathbb E |H_{ij}|^2$ form a band matrix with typical band width $1ll Wll N$. We consider the generalized resolvent of $H$ defined as $G(Z):=(H - Z)^{-1}$, where $Z$ is a deterministic diagonal matrix such that $Z_{ij}=left(z 1_{1leq i leq W}+widetilde z 1_{ i > W} right) delta_{ij}$, with two distinct spectral parameters $zin mathbb C_+:={zin mathbb C:{rm Im} z>0}$ and $widetilde zin mathbb C_+cup mathbb R$. In this paper, we prove a sharp bound for the local law of the generalized resolvent $G$ for $Wgg N^{3/4}$. This bound is a key input for the proof of delocalization and bulk universality of random band matrices in cite{PartI}. Our proof depends on a fluctuations averaging bound on certain averages of polynomials in the resolvent entries, which will be proved in cite{PartIII}.
We introduce a Gibbs measure on nearest-neighbour paths of length $t$ in the Euclidean $d$-dimensional lattice, where each path is penalised by a factor proportional to the size of its boundary and an inverse temperature $beta$. We prove that, for all $beta>0$, the random walk condensates to a set of diameter $(t/beta)^{1/3}$ in dimension $d=2$, up to a multiplicative constant. In all dimensions $dge 3$, we also prove that the volume is bounded above by $(t/beta)^{d/(d+1)}$ and the diameter is bounded below by $(t/beta)^{1/(d+1)}$. Similar results hold for a random walk conditioned to have local time greater than $beta$ everywhere in its range when $beta$ is larger than some explicit constant, which in dimension two is the logarithm of the connective constant.
In this paper we study the metastable behavior of one of the simplest disordered spin system, the random field Curie-Weiss model. We will show how the potential theoretic approach can be used to prove sharp estimates on capacities and metastable exit times also in the case when the distribution of the random field is continuous. Previous work was restricted to the case when the random field takes only finitely many values, which allowed the reduction to a finite dimensional problem using lumping techniques. Here we produce the first genuine sharp estimates in a context where entropy is important.
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