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Register a new userConformal invariance of double random currents and the XOR-Ising model I: identification of the limit
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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.
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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.
In a recent paper [15], Giardin{`a}, Giberti, Hofstad, Prioriello have proved a law of large number and a central limit theorem with respect to the annealed measure for the magnetization of the Ising model on some random graphs including the random 2-regular graph. We present a new proof of their results, which applies to all random regular graphs. In addition, we prove the existence of annealed pressure in the case of configuration model random graphs.
We consider branching random walks in $d$-dimensional integer lattice with time-space i.i.d. offspring distributions. When $d ge 3$ and the fluctuation of the environment is well moderated by the random walk, we prove a central limit theorem for the density of the population, together with upper bounds for the density of the most populated site and the replica overlap. We also discuss the phase transition of this model in connection with directed polymers in random environment.
Tensor models generalize matrix models and generate colored triangulations of pseudo-manifolds in dimensions $Dgeq 3$. The free energies of some models have been recently shown to admit a double scaling limit, i.e. large tensor size $N$ while tuning to criticality, which turns out to be summable in dimension less than six. This double scaling limit is here extended to arbitrary models. This is done by means of the Schwinger--Dyson equations, which generalize the loop equations of random matrix models, coupled to a double scale analysis of the cumulants.
We study continuous-time (variable speed) random walks in random environments on $mathbb{Z}^d$, $dge2$, where, at time $t$, the walk at $x$ jumps across edge $(x,y)$ at time-dependent rate $a_t(x,y)$. The rates, which we assume stationary and ergodic with respect to space-time shifts, are symmetric and bounded but possibly degenerate in the sense that the total jump rate from a vertex may vanish over finite intervals of time. We formulate conditions on the environment under which the law of diffusively-scaled random walk path tends to Brownian motion for almost every sample of the rates. The proofs invoke Moser iteration to prove sublinearity of the corrector in pointwise sense; a key additional input is a conversion of certain weighted energy norms to ordinary ones. Our conclusions apply to random walks on dynamical bond percolation and interacting particle systems as well as to random walks arising from the Helffer-Sjostrand representation of gradient models with certain non-strictly convex potentials.
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