The immediate purpose of the paper was neither to review the basic definitions of percolation theory nor to rehearse the general physical notions of universality and renormalization (an important technique to be described in Part Two). It was rather to describe as concretely as possible, although in hypothetical form, the geometric aspects of universality, especially conformal invariance, in the context of percolation, and to present the numerical results that support the hypotheses. On the other hand, one ulterior purpose is to draw the attention of mathematicians to the mathematical problems posed by the physical notions. Some precise basic definitions are necessary simply to orient the reader. Moreover a brief description of scaling and universality on the one hand and of renormalization on the other is also essential in order to establish their physical importance and to clarify their mathematical content.
The smallest deformation of the minimal model M(2,3) that can accommodate Cardy's derivation of the percolation crossing probability is presented. It is shown that this leads to a consistent logarithmic conformal field theory at c=0. A simple recipe for computing the associated fusion rules is given. The differences between this theory and the other recently proposed c=0 logarithmic conformal field theories are underlined. The discussion also emphasises the existence of invariant logarithmic couplings that generalise Gurarie's anomaly number.
We consider the simple random walk on the (unique) infinite cluster of super-critical bond percolation in $\Z^d$ with $d\ge2$. We prove that, for almost every percolation configuration, the path distribution of the walk converges weakly to that of non-degenerate, isotropic Brownian motion. Our analysis is based on the consideration of a harmonic deformation of the infinite cluster on which the random walk becomes a square-integrable martingale. The size of the deformation, expressed by the so called corrector, is estimated by means of ergodicity arguments.
These lectures give an introduction to the methods of conformal field theory as applied to deriving certain results in two-dimensional critical percolation: namely the probability that there exists at least one cluster connecting two disjoint segments of the boundary of a simply connected region; and the mean number of such clusters. No previous familiarity with conformal field theory is assumed, but in the course of the argument many of its important concepts are introduced in as simple a manner as possible. A brief account is also given of some recent alternative approaches to deriving these…
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…
The conformal bootstrap is applied to percolation and dilute self-avoiding polymers, two theories with Virasoro central charge $c=0$ in two dimensions. In both cases we propose a spectrum of operators motivated by Virasoro symmetry which is devoid of a stress energy tensor as an approximate means of enforcing $c=0$. Percolation is treated in $2\leq D \leq 6$ dimensions, and the self-avoiding walk in $2 \leq D \leq 4$.
We study scaling limits and conformal invariance of critical site percolation on triangular lattice. We show that some percolation-related quantities are harmonic conformal invariants, and calculate their values in the scaling limit. As a particular case we obtain conformal invariance of the crossing probabilities and Cardy's formula. Then we prove existence, uniqueness, and conformal invariance of the continuum scaling limit.
We show that for critical site percolation on the triangular lattice two new observables have conformally invariant scaling limits. In particular the expected number of clusters separating two pairs of points converges to an explicit conformal invariant. Our proof is independent of earlier results and $SLE$ techniques, and in principle should provide a new approach to establishing conformal invariance of percolation.
We study four-point functions of critical percolation in two dimensions, and more generally of the Potts model. We propose an exact ansatz for the spectrum: an infinite, discrete and non-diagonal combination of representations of the Virasoro algebra. Based on this ansatz, we compute four-point functions using a numerical conformal bootstrap approach. The results agree with Monte-Carlo computations of connectivities of random clusters.
We consider critical percolation on the triangular lattice in a bounded simply connected domain with boundary conditions that force an interface between two prescribed boundary points. We say the interface forms a "near-loop" when it comes within one lattice spacing of itself. We define a new curve by erasing these near-loops as we traverse the interface. Our Monte Carlo simulations of this model lead us to conclude that the scaling limit of this loop-erased percolation interface is conformally invariant and has fractal dimension 4/3. However, it is not SLE_8/3. We also consider the process in…
The relations and differences between various classification problems arising in the context of local two-dimensional conformal QFT, modular invariants, and subfactors are discussed. The extent to which locality implies modular invariance, is exhibited.
Using conformal field theory, we derive several new crossing formulas at the two-dimensional percolation point. High-precision simulation confirms these results. Integrating them gives a unified derivation of Cardy's formula for the horizontal crossing probability $Π_h(r)$, Watts' formula for the horizontal-vertical crossing probability $Π_{hv}(r)$, and Cardy's formula for the expected number of clusters crossing horizontally $\mathcal{N}_h(r)$. The main step in our approach implies the identification of the derivative of one primary operator with another. We present operator identities that s…
The bulk one-arm exponent for the CLE$_{κ'}$ percolations
Peking University（北京大学）；Beijing International Center for Mathematical Research, Peking University（北京大学北京国际数学研究中心）；Courant Institute of Mathematical Sciences, New York University（纽约大学柯朗数学科学研究所）；University of Pennsylvania（宾夕法尼亚大学）
We construct discrete holomorphic observables in the Ising model at criticality and show that they have conformally covariant scaling limits (as mesh of the lattice tends to zero). In the sequel those observables are used to construct conformally invariant scaling limits of interfaces. Though Ising model is often cited as a classical example of conformal invariance, it seems that ours is the first paper where it is actually established.
We show that if the three dimensional self-avoiding walk (SAW) is conformally invariant, then one can compute the hitting densities for the SAW in a half space and in a sphere. We test these predictions by Monte Carlo simulations and find excellent agreement, thus providing evidence that the SAW is conformally invariant in three dimensions.
Recent developments on emergence of logarithmic terms in correlators or response functions of models which exhibit dynamical symmetries analogous to conformal invariance in not necessarily relativistic systems are reviewed. The main examples of these are logarithmic Schrödinger-invariance and logarithmic conformal Galilean invariance. Some applications of these ideas to statistical physics are described.
By use of conformal field theory, we discover several exact factorizations of higher-order density correlation functions in critical two-dimensional percolation. Our formulas are valid in the upper half-plane, or any conformally equivalent region. We find excellent agreement of our results with high-precision computer simulations. There are indications that our formulas hold more generally.
In this note, we illustrate how the two-dimensional theory of elasticity provides a physical example of field theory displaying scale but not conformal invariance.
The covariant two-point functions, derived from Ward identities in direct space, can be affected by consistency problems and can become unbounded for large time- or space-separations. This difficulty arises for several extensions of dynamical scaling, for example Schrödinger-invariance, conformal Galilei invariance or meta-conformal invariance, but not for standard ortho-conformal invariance. For meta-conformal invariance in 1+1 dimensions, these difficulties can be cured by going over to a dual space and an extension of these dynamical symmetries through the construction of a new generator in…
Building on an analogy with conformal invariance, local scale transformations consistent with dynamical scaling are constructed. Two types of local scale invariance are found which act as dynamical space-time symmetries of certain non-local free field theories. Physical applications include uniaxial Lifshitz points and ageing in simple ferromagnets.
Invariance under finite conformal transformations in Minkowski space and the Wightman axioms imply strong locality (Huygens principle) and rationality of correlation functions, thus providing an extension of the concept of vertex algebra to higher dimensions. Gibbs (finite temperature) expectation values appear as elliptic functions in the conformal time. We survey and further pursue our program of constructing a globally conformal invariant model of a hermitean scalar field L of scale dimension four in Minkowski space-time which can be interpreted as the Lagrangian density of a gauge field th…
Many 2D lattice models of physical phenomena are conjectured to have conformally invariant scaling limits: percolation, Ising model, self-avoiding polymers, ... This has led to numerous exact (but non-rigorous) predictions of their scaling exponents and dimensions. We will discuss how to prove the conformal invariance conjectures, especially in relation to Schramm-Loewner Evolution.
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.
It is widely believed that the celebrated 2D Ising model at criticality has a universal and conformally invariant scaling limit, which is used in deriving many of its properties. However, no mathematical proof of universality and conformal invariance has ever been given, and even physics arguments support (a priori weaker) Möbius invariance. We introduce discrete holomorphic fermions for the 2D Ising model at criticality on a large family of planar graphs. We show that on bounded domains with appropriate boundary conditions, those have universal and conformally invariant scaling limits, thus p…
Building on an analogy with conformal invariance, local scale transformations consistent with dynamical scaling are constructed. Two types of local scale invariance are found which act as dynamical space-time symmetries of certain non-local free field theories. The scaling form of two-point functions is completely fixed by the requirement of local scale invariance. These predictions are confirmed through tests in the 3D ANNNI model at its Lifshitz point and in ageing phenomena of simple ferromagnets, here studied through the kinetic Ising model with Glauber dynamics.
We present experimental evidence of statistical conformal invariance in isocontours of fluid thickness in experiments of two-dimensional turbulence using soap films. A Schlieren technique is used to visualize regions of the flow with constant film thickness, and association of isocontours with Schramm-Löwner evolution (SLE) is used to identify conformal invariance. In experiments where an inverse energy cascade develops, statistical evidence is consistent with such an association. The diffusivity of the associated one-dimensional Brownian process is close to 8/3, a value previously identified …
Dynamical symmetries of the collisionless Boltzmann transport equation, or Vlasov equation, but under the influence of an external driving force, are derived from non-standard representations of the $2D$ conformal algebra. In the case without external forces, the symmetry of the conformally invariant transport equation is first generalised by considering the particle momentum as an independent variables. This new conformal representation can be further extended to include an external force. The construction and possible physical applications are outlined.
Following the approach outlined in [26], convergence to SLE$_6$ of the Exploration Processes for the correlated bond-triangular type models studied in [11] is established. This puts the said models in the same universality class as the standard site percolation model on the triangular lattice [27]. In the context of these models, the result is proven for all domains with boundary Minkowski dimension less than two. Moreover, the proof of convergence applies in the context of general critical 2D percolation models and for general domains, under the stipulation that Cardy's Formula can be establi…
The new manifestation of conformal invariance for a massless scalar particle in a Riemannian spacetime of general relativity is found. Conformal transformations conserve the Hamiltonian and wave function in the Foldy-Wouthuysen representation. Similarity of manifestations of conformal invariance for massless scalar and Dirac particles is proved. New exact Foldy-Wouthuysen Hamiltonians are derived for both massive and massless scalar particles in a general static spacetime and in a frame rotating in the Kerr field approximated by a spatially isotropic metric. The latter case covers an observer …
Using the planar Temperley-Lieb algebra, critical bond percolation on the square lattice is incorporated as ${\cal LM}(2,3)$ in the family of Yang-Baxter integrable logarithmic minimal models ${\cal LM}(p,p')$. We consider this model in the presence of boundaries and with periodic boundary conditions. Inspired by Kuniba, Sakai and Suzuki, we rewrite the recently obtained infinite $Y$-system of functional equations. We obtain nonlinear integral equations in the form of a closed finite set of TBA equations described by a $D_3$ Dynkin diagram. Following the methods of Klümper and Pearce, we solve…
We use SLE(6) paths to construct a process of continuum nonsimple loops in the plane and prove that this process coincides with the full continuum scaling limit of 2D critical site percolation on the triangular lattice -- that is, the scaling limit of the set of all interfaces between different clusters. Some properties of the loop process, including conformal invariance, are also proved.
Conformal loop ensembles are random collections of loops in a simply connected domain, whose laws are characterized by a natural conformal invariance property. The set of points not surrounded by any CLE loop is a natural random and conformally invariant analog of the Sierpinski gasket or carpet. In the present paper, we derive a direct relationship between each CLE consisting of simple disjoint loops (CLE($κ$) with $κ$ between 8/3 and 4) and the corresponding CLE($κ'$) where $κ':=16/κ$, a CLE consisting of non-disjoint loops. This is the continuum analog of the Edwards-Sokal coupling (between…
Crossing probabilities for critical 2-D percolation on large but finite lattices have been derived via boundary conformal field theory. These predictions agree very well with numerical results. However, their derivation is heuristic and there is evidence of additional symmetries in the problem. This contribution gives a preliminary examination some unusual modular behavior of these quantities. In particular, the derivatives of the "horizontal" and "horizontal-vertical" crossing probabilities transform as a vector modular form, one component of which is an ordinary modular form and the other th…
General Relativity receives quantum corrections relevant at cosmological distance scales from the conformal scalar degrees of freedom required by the trace anomaly of the quantum stress tensor in curved space. In the theory including the trace anomaly terms, the cosmological "constant" becomes dynamical and hence potentially dependent upon both space and time. The fluctuations of these anomaly scalars may also influence the spectrum and statistics of the Cosmic Microwave Background. Under the hypothesis that scale invariance should be promoted to full conformal invariance, an hypothesis suppor…
The behaviour of the 3D axial next-nearest neighbour Ising (ANNNI) model at the uniaxial Lifshitz point is studied using Monte Carlo techniques. A new variant of the Wolff cluster algorithm permits the analysis of systems far larger than in previous studies. The Lifshitz point critical exponents are $α=0.18(2)$, $β=0.238(5)$ and $γ=1.36(3)$. Data for the spin-spin correlation function are shown to be consistent with the explicit scaling function derived from the assumption of local scale invariance, which is a generalization of conformal invariance to the anisotropic scaling {\em at} the Lifsh…
