Rotational invariance in critical planar lattice models
University of Fribourg（弗里堡大学）；Institut des Hautes Études Scientifiques（高等科学研究所）；Université Paris-Saclay（巴黎萨克雷大学）；Université de Genève（日内瓦大学）；ENS Lyon（里昂高等师范学院）；Princeton University（普林斯顿大学）
Hugo Duminil-Copin, Karol Kajetan Kozlowski, Dmitry Krachun, Ioan Manolescu, Mendes Oulamara
备注：
This new version of the paper substantially simplifies and streamlines the proof. In particular, it does not make use of Bethe ansatz computations to deduce that the drift of the transformations is null. It also contains improvements of the results: quantitative bounds and uniformity in angles for the universality theorem
We prove that the large-scale properties of a number of two-dimensional lattice models are rotationally invariant. More precisely, we prove that the random-cluster model on the square lattice with cluster-weight $1\le q\le 4$ exhibits rotational invariance at large scales. This covers the case of Bernoulli percolation on the square lattice as an important example. We deduce that the correlations of the critical Potts models with $q\in\{2,3,4\}$ colours are rotationally invariant at large scales.
Our result is instrumental in proving the convergence of the six-vertex model to the Gaussian Free Field in a separate paper.
Conformal invariance of the planar four-state Potts model and its random-cluster representation
CNRS, Sorbonne Université, LPSM（法国国家科学研究中心，索邦大学，LPSM）；Université de Fribourg（弗里堡大学）；Université de Genève（日内瓦大学）；Institut des Hautes Études Scientifiques（高等科学研究所）
A number of recent new proofs of Euler's celebrated identity are presented, all of which are probabilistic in nature and depend on the conformal invariance of planar Brownian motion.
We rigorously prove the existence and the conformal invariance of scaling limits of the magnetization and multi-point spin correlations in the critical Ising model on arbitrary simply connected planar domains. This solves a number of conjectures coming from the physical and the mathematical literature. The proof relies on convergence results for discrete holomorphic spinor observables and probabilistic techniques.
These lecture notes provide a (almost) self-contained account on conformal invariance of the planar critical Ising and FK-Ising models. They present the theory of discrete holomorphic functions and its applications to planar statistical physics (more precisely to the convergence of fermionic observables). Convergence to SLE is discussed briefly. Many open questions are included.
In this review paper, we first discuss some open problems related to two-dimensional self-avoiding paths and critical percolation. We then review some closely related results (joint work with Greg Lawler and Oded Schramm) on critical exponents for two-dimensional simple random walks, Brownian motions and other conformally invariant random objects.
The dimer model is the study of random dimer covers (perfect matchings) of a graph. A double-dimer configuration on a graph $G$ is a union of two dimer covers of $G$. We introduce quaternion weights in the dimer model and show how they can be used to study the homotopy classes (relative to a fixed set of faces) of loops in the double dimer model on a planar graph. As an application we prove that, in the scaling limit of the "uniform" double-dimer model on ${\mathbb Z}^2$ (or on any other bipartite planar graph conformally approximating $\mathbb C$), the loops are conformally invariant. As othe…
We derive a local limit theorem for normal, moderate, and large deviations for symmetric simple random walk on the square lattice in dimensions one and two that is an improvement of existing results for points that are particularly distant from the walk's starting point. More specifically, we give explicit asymptotic expressions in terms of $n$ and $x$, where $x$ is thought of as dependent on $n$, in dimensions one and two for $P(S_n=x)$, the probability that symmetric simple random walk $S$ started at the origin is at some point $x$ at time $n$, that are valid for all $x$. We also show that t…
Schramm Loewner Evolution (SLE) is a one-parameter family of random planar curves introduced by Oded Schramm in 1999 as the candidates for the scaling limits of the interfaces in the planar critical lattice models. This is the only possible process with conformal invariance and a certain "domain Markov property". In 2010, Chelkak and Smirnov proved the conformal invariance of the scaling limits of the critial planar FK-Ising model which gave the convergence of the interface to SLE$_{16/3}$. We derive the arm exponents of SLE$_κ$ for $κ\in (4,8)$. Combining with the convergence of the interface…
We prove that the scaling limit of loop-erased random walk in a simply connected domain $D$ is equal to the radial SLE(2) path in $D$. In particular, the limit exists and is conformally invariant. It follows that the scaling limit of the uniform spanning tree in a Jordan domain exists and is conformally invariant. Assuming that the boundary of the domain is a $C^1$ simple closed curve, the same method is applied to show that the scaling limit of the uniform spanning tree Peano curve, where the tree is wired along a proper arc $A$ on the boundary, is the chordal SLE(8) path in the closure of $D…
There are many classical random walk in random environment results that apply to ergodic random planar environments. We extend some of these results to random environments in which the length scale varies from place to place, so that the law of the environment is in a certain sense only translation invariant {\em modulo scaling}. For our purposes, an ``environment'' consists of an infinite random planar map embedded in $\mathbb C$, each of whose edges comes with a positive real conductance. Our main result is that under modest constraints (translation invariance modulo scaling together with th…
For a symmetric random walk in $Z^2$ with $2+δ$ moments, we represent $|\mathcal{R}(n)|$, the cardinality of the range, in terms of an expansion involving the renormalized intersection local times of a Brownian motion. We show that for each $k\geq 1$ \[ (\log n)^k [ \frac{1}{n} |\mathcal{R}(n)| +\sum_{j=1}^k (-1)^j (\textstyle{\frac1{2π}}\log n +c_X)^{-j} γ_{j,n}]\to 0, \qquad a.s. \] where $W_t$ is a Brownian motion, $W^{(n)}_t=W_{nt}/\sqrt n$, $γ_{j,n}$ is the renormalized intersection local time at time 1 for $W^{(n)}$, and $c_X$ is a constant depending on the distribution of the random wal…
This is an introductory account of the emergence of conformal invariance in the scaling limit of planar critical percolation. We give an exposition of Smirnov's theorem (2001) on the conformal invariance of crossing probabilities in site percolation on the triangular lattice. We also give an introductory account of Schramm-Loewner evolutions (SLE(k)), a one-parameter family of conformally invariant random curves discovered by Schramm (2000). The article is organized around the aim of proving the result, due to Smirnov (2001) and to Camia and Newman (2007), that the percolation exploration path…
Random planar maps are considered in the physics literature as the discrete counterpart of random surfaces. It is conjectured that properly rescaled random planar maps, when conditioned to have a large number of faces, should converge to a limiting surface whose law does not depend, up to scaling factors, on details of the class of maps that are sampled. Previous works on the topic, starting with Chassaing and Schaeffer, have shown that the radius of a random quadrangulation with $n$ faces, that is, the maximal graph distance on such a quadrangulation to a fixed reference point, converges in d…
This paper describes joint work with Oded Schramm and Wendelin Werner establishing the values of the planar Brownian intersection exponents from which one derives the Hausdorff dimension of certain exceptional sets of planar Brownian motion. In particular, we proof a conjecture of Mandelbrot that the dimension of the frontier is 4/3. The proof uses a universality principle for conformally invariant measures and a new process, the stochastic Loewner evolution ($SLE$), introduced by Schramm. These ideas can be used to study other planar lattice models from statistical physics at criticality. I d…
We introduce Kadanoff-Ceva order-disorder operators in the quantum Ising model. This approach was first used for the classical planar Ising model and recently put back to the stage. This representation turns out to be equivalent to the loop expansion of Sminorv's fermionic observables and is particularly interesting due to its simple and compact formulation. Using this approach, we are able to extend different results known in the classical planar Ising model, such as the conformal invariance/covariance of correlations and the energy-density, to the spin-representation of the (1+1) dimensional…
For a broad class of planar Markov processes, viz. Lévy processes satisfying certain conditions (valid \textit{eg} in the case of Brownian motion and Lévy flights), we establish an exact, universal formula describing the shape of the convex hull of sample paths. We show indeed that the average number of edges joining paths' points separated by a time-lapse $Δτ\in \left[Δτ_1, Δτ_2\right]$ is equal to $2\ln \left(Δτ_2 / Δτ_1 \right)$, regardless of the specific distribution of the process's increments and regardless of its total duration $T$. The formula also exhibits invariance when the time sc…
In this essay, we briefly discuss recent developments, started a decade ago in the seminal work of Smirnov and continued by a number of authors, centered around the conformal invariance of the critical planar Ising model on $\mathbb{Z}^2$ and, more generally, of the critical Z-invariant Ising model on isoradial graphs (rhombic lattices). We also introduce a new class of embeddings of general weighted planar graphs (s-embeddings), which might, in particular, pave the way to true universality results for the planar Ising model.
We prove that crossing probabilities for the critical planar Ising model with free boundary conditions are conformally invariant in the scaling limit, a phenomenon first investigated numerically by Langlands, Lewis and Saint-Aubin. We do so by establishing the convergence of certain exploration processes towards SLE$(3,\frac{-3}2,\frac{-3}2)$. We also construct an exploration tree for free boundary conditions, analogous to the one introduced by Sheffield.
We prove an almost sure invariance principle (approximation by d-dimensional Brownian motion) for vector-valued Holder observables of large classes of nonuniformly hyperbolic dynamical systems. These systems include Axiom~A diffeomorphisms and flows as well as systems modelled by Young towers with moderate tail decay rates. In particular, the position variable of the planar periodic Lorentz gas with finite horizon approximates a 2-dimensional Brownian motion.
