Critical phenomena and conformal invariance in the plane
BMETE95MM23, Spring 2012
List of suggested papers for presentation

Lecturer: Gábor Pete

You are welcome to ask my help when reading the paper. Aim at understanding what your chosen paper is about, and at each of you presenting at least one non-trivial proof. You can use the paper or your notes for the presentation, but make sure you know the basic definitions and ideas. Duration should be 60-90 minutes for a pair, and around 45 minutes for a single presenter.

Conformal invariance and winding numbers of planar Brownian motion
Section 7.2 of the book Brownian motion by Peter Mörters and Yuval Peres.
8 pages.

On monochromatic arm exponents for 2D critical percolation
Vincent Beffara, Pierre Nolin
18 pages, 4 figures
We investigate the so-called "monochromatic arm exponents" for critical percolation in two dimensions. These exponents, describing the probability of observing j disjoint macroscopic paths, are shown to exist and to form a different family from the (now well understood) polychromatic exponents. More specifically, our main result is that the monochromatic j-arm exponent is strictly between the polychromatic j-arm and (j+1)-arm exponents.

One-arm exponent for critical 2D percolation
Gregory F. Lawler, Oded Schramm, Wendelin Werner
18 pages, 1 figure
The probability that the cluster of the origin in critical site percolation on the triangular grid has diameter larger than $R$ is proved to decay like $R^{-5/48}$ as $R\to\infty$.

Quantitative noise sensitivity and exceptional times for percolation
Oded Schramm, Jeffrey E. Steif
64 pages
One goal of this paper is to prove that dynamical critical site percolation on the planar triangular lattice has exceptional times at which percolation occurs. In doing so, new quantitative noise sensitivity results for percolation are obtained. The latter is based on a novel method for controlling the "level k" Fourier coefficients via the construction of a randomized algorithm which looks at random bits, outputs the value of a particular function but looks at any fixed input bit with low probability. We also obtain upper and lower bounds on the Hausdorff dimension of the set of percolating times. We then study the problem of exceptional times for certain "k-arm" events on wedges and cones. As a corollary of this analysis, we prove, among other things, that there are no times at which there are two infinite "white" clusters, obtain an upper bound on the Hausdorff dimension of the set of times at which there are both an infinite white cluster and an infinite black cluster and prove that for dynamical critical bond percolation on the square grid there are no exceptional times at which three disjoint infinite clusters are present.
(A follow-up is The Fourier spectrum of critical percolation by Garban, Pete and Schramm.)

The self-dual point of the two-dimensional random-cluster model is critical for $q\geq 1$
Vincent Beffara, Hugo Duminil-Copin
27 pages, 10 figures
We prove a long-standing conjecture on random-cluster models, namely that the critical point for such models with parameter $q\geq1$ on the square lattice is equal to the self-dual point $p_{sd}(q) = \sqrt q /(1+\sqrt q)$. This gives a proof that the critical temperature of the $q$-state Potts model is equal to $\log (1+\sqrt q)$ for all $q\geq 2$. We further prove that the transition is sharp, meaning that there is exponential decay of correlations in the sub-critical phase. The techniques of this paper are rigorous and valid for all $q\geq 1$, in contrast to earlier methods valid only for certain given $q$. The proof extends to the triangular and the hexagonal lattices as well.

An introduction to the dimer model [Szanto Bandi + Rozsa Levente + Vajna Szabolcs]
Richard Kenyon
Lecture notes from a minicourse given at the ICTP in May 2002.
See also
Lectures on Dimers
Richard Kenyon
These are lecture notes for lectures at the Park City Math Institute, summer 2007. We cover aspects of the dimer model on planar, periodic bipartite graphs, including local statistics, limit shapes and fluctuations.

Discrete complex analysis on isoradial graphs [Nagy Attila + ?]
Dmitry Chelkak, Stanislav Smirnov
35 pages, 4 figures.
We study discrete complex analysis and potential theory on a large family of planar graphs, the so-called isoradial ones. Along with discrete analogues of several classical results, we prove uniform convergence of discrete harmonic measures, Green's functions and Poisson kernels to their continuous counterparts. Among other applications, the results can be used to establish universality of the critical Ising and other lattice models.

Unfortunately, on the last class I didn't have time to define Smirnov's fermionic observable, which is more-or-less the discrete holomorphic function for the FK-Ising model whose convergence to a true holomorphic function is responsible for the conformal invariance of the model. But either of the next two papers will give you the main idea. (The convergence is much harder to prove than in the percolation case, that's why I'm not suggesting Smirnov's Ann of Math paper Conformal invariance in random-cluster models I. Holomorphic fermions in the Ising model for presentation.)

Connection probabilities and RSW-type bounds for the FK Ising model
Hugo Duminil-Copin, Clément Hongler, Pierre Nolin
31 pages, 9 figures
We prove Russo-Seymour-Welsh-type uniform bounds on crossing probabilities for the FK Ising model at criticality, independent of the boundary conditions. Our proof relies mainly on Smirnov's fermionic observable for the FK Ising model, which allows us to get precise estimates on boundary connection probabilities. It remains purely discrete, in particular we do not make use of any continuum limit, and it can be used to derive directly several noteworthy results - some new and some not - among which the fact that there is no spontaneous magnetization at criticality, tightness properties for the interfaces, and the existence of several critical exponents, in particular the half-plane one-arm exponent.

Smirnov's fermionic observable away from criticality
Vincent Beffara, Hugo Duminil-Copin
19 pages, 6 figures
In a recent and celebrated article, Smirnov defines an observable for the self-dual random-cluster model with cluster weight $q=2$ on the square lattice $\Z^2$, and uses it to obtain conformal invariance in the scaling limit. We study this observable away from the self-dual point. From this, we obtain a new derivation of the fact that the self-dual and critical points coincide, which implies that the critical inverse temperature of the Ising model equals $\frac12\log(1+\sqrt2)$. Moreover, we relate the correlation length of the model to the large deviation behavior of a certain massive random walk (thus confirming an observation by Messikh), which allows us to compute it explicitly.
(A follow-up is The near-critical planar FK-Ising model by Duminil-Copin, Garban and Pete.)

The connective constant of the honeycomb lattice equals $\sqrt{2+\sqrt2}$
Hugo Duminil-Copin, Stanislav Smirnov
11 pages, 4 figures
We provide the first mathematical proof that the connective constant of the hexagonal lattice is equal to $\sqrt{2+\sqrt 2}$. This value has been derived non rigorously by B. Nienhuis in 1982, using Coulomb gas approach from theoretical physics. Our proof uses a parafermionic observable for the self avoiding walk, which satisfies a half of the discrete Cauchy-Riemann relations. Establishing the other half of the relations (which conjecturally holds in the scaling limit) would also imply convergence of the self-avoiding walk to SLE(8/3).

Critical Ising on the square lattice mixes in polynomial time
Eyal Lubetzky, Allan Sly
26 pages; 5 figures
The Ising model is widely regarded as the most studied model of spin-systems in statistical physics. The focus of this paper is its dynamic (stochastic) version, the Glauber dynamics, introduced in 1963 and by now the most popular means of sampling the Ising measure. Intensive study throughout the last three decades has yielded a rigorous understanding of the spectral-gap of the dynamics on $\Z^2$ everywhere except at criticality. While the critical behavior of the Ising model has long been the focus for physicists, mathematicians have only recently developed an understanding of its critical geometry with the advent of SLE, CLE and new tools to study conformally invariant systems.
A rich interplay exists between the static and dynamic models. At the static phase-transition for Ising, the dynamics is conjectured to undergo a critical slowdown: At high temperature the inverse-gap is O(1), at the critical $\beta_c$ it is polynomial in the side-length and at low temperature it is exponential in it. A seminal series of papers verified this on $\Z^2$ except at $\beta=\beta_c$ where the behavior remained a challenging open problem.
Here we establish the first rigorous polynomial upper bound for the critical mixing, thus confirming the critical slowdown for the Ising model in $\Z^2$. Namely, we show that on a finite box with arbitrary (e.g. fixed, free, periodic) boundary conditions, the inverse-gap at $\beta=\beta_c$ is polynomial in the side-length. The proof harnesses recent understanding of the scaling limit of critical Fortuin-Kasteleyn representation of the Ising model together with classical tools from the analysis of Markov chains.

Gaussian free fields for mathematicians
Scott Sheffield
27 pages, 1 figure.
The d-dimensional Gaussian free field (GFF), also called the (Euclidean bosonic) massless free field, is a d-dimensional-time analog of Brownian motion. Just as Brownian motion is the limit of the simple random walk (when time and space are appropriately scaled), the GFF is the limit of many incrementally varying random functions on d-dimensional grids. We present an overview of the GFF and some of the properties that are useful in light of recent connections between the GFF and the Schramm-Loewner evolution.

The harmonic explorer and its convergence to SLE(4)
Oded Schramm, Scott Sheffield
The harmonic explorer is a random grid path. Very roughly, at each step the harmonic explorer takes a turn to the right with probability equal to the discrete harmonic measure of the left-hand side of the path from a point near the end of the current path. We prove that the harmonic explorer converges to SLE(4) as the grid gets finer.
(This is a prequel to Contour lines of the two-dimensional discrete Gaussian free field by Schramm and Sheffield. An even more serious continuation is Imaginary Geometry I: Interacting SLEs by Miller and Sheffield, which (together with parts II and III) are part of the Liouville Quantum Gravity program in the next item.)

Liouville Quantum Gravity and KPZ
Bertrand Duplantier, Scott Sheffield
56 pages.
Consider a bounded planar domain D, an instance h of the Gaussian free field on D (with Dirichlet energy normalized by 1/(2\pi)), and a constant 0 < gamma < 2. The Liouville quantum gravity measure on D is the weak limit as epsilon tends to 0 of the measures \epsilon^{\gamma^2/2} e^{\gamma h_\epsilon(z)}dz, where dz is Lebesgue measure on D and h_\epsilon(z) denotes the mean value of h on the circle of radius epsilon centered at z. Given a random (or deterministic) subset X of D one can define the scaling dimension of X using either Lebesgue measure or this random measure. We derive a general quadratic relation between these two dimensions, which we view as a probabilistic formulation of the KPZ relation from conformal field theory. We also present a boundary analog of KPZ (for subsets of the boundary of D). We discuss the connection between discrete and continuum quantum gravity and provide a framework for understanding Euclidean scaling exponents via quantum gravity.
(Sequels are Conformal weldings of random surfaces: SLE and the quantum gravity zipper and Quantum gravity and inventory accumulation by Sheffield.)
See also:
A very nice introduction to this topic is a recent Bourbaki lecture by Christophe Garban, Quantum gravity and the KPZ formula [after Duplantier-Sheffield].

The Uniform Infinite Planar Quadrangulation [Koi Tomi + Horvath Illes, June 7]
Nicolas Curien
18 pages, lecture notes for YEP IX, 2012.
Uniform random planar maps are discrete analogues of what should be the random planar metric space underlying Liouville Quantum Gravity. These notes are concerned with the local limit of the uniform planar quadrangulation with $n$ faces, and give a transparent proof that the volume growth of this infinite planar graph is of degree 4.