Critical percolation exploration path and SLE 6: a proof of convergence

It was argued by Schramm and Smirnov that the critical site percolation exploration path on the triangular lattice converges in distribution to the trace of chordal SLE ₆. We provide here a detailed proof, which relies on Smirnov’s theorem that crossing probabilities have a conformally invariant scaling limit (given by Cardy’s formula). The version of convergence to SLE ₆ that we prove suffices for the Smirnov–Werner derivation of certain critical percolation crossing exponents and for our analysis of the critical percolation full scaling limit as a process of continuum nonsimple loops. Research of Charles M.Newman was partially supported by the US NSF under grants DMS-01-04278 and DMS-06-06696.     

Continuum percolation with steps in the square or the disc

In 1961 Gilbert defined a model of continuum percolation in which points are placed in the plane according to a Poisson process of density 1, and two are joined if one lies within a disc of area A about the other. We prove some good bounds on the critical area A_c for percolation in this model. The proof is in two parts: First we give a rigorous reduction of the problem to a finite problem, and then we solve this problem using Monte‐Carlo methods. We prove that, with 99.99% confidence, the critical area lies between 4.508 and 4.515. For the corresponding problem with the disc replaced by the square we prove, again with 99.99% confidence, that the critical area lies between 4.392 and 4.398. © 2005 Wiley Periodicals, Inc. Random Struct. Alg., 2005     

Universality in two‐dimensional enhancement percolation

We consider a type of dependent percolation introduced in 2 , where it is shown that certain “enhancements” of independent (Bernoulli) percolation, called essential, make the percolation critical probability strictly smaller. In this study we first prove that, for two‐dimensional enhancements with a natural monotonicity property, being essential is also a necessary condition to shift the critical point. We then show that (some) critical exponents and the scaling limit of crossing probabilities of a two‐dimensional percolation process are unchanged if the process is subjected to a monotonic enhancement that is not essential. This proves a form of universality for all dependent percolation models obtained via a monotonic enhancement (of Bernoulli percolation) that does not shift the critical point. For the case of site percolation on the triangular lattice, we also prove a stronger form of universality by showing that the full scaling limit 12 , 13 is not affected by any monotonic enhancement that does not shift the critical point. © 2008 Wiley Periodicals, Inc. Random Struct. Alg., 2008     

Critical percolation: the expected number of clusters in a rectangle

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 might provide a new approach to establishing conformal invariance of percolation.     

Continuum limits for classical sequential growth models

A random graph order, also known as a transitive percolation process, is defined by taking a random graph on the vertex set {0,…,n ? 1} and putting i below j if there is a path i = i₁…i_k = j in the graph with i₁ < … < i_k. Rideout and Sorkin 14 provide computational evidence that suitably normalized sequences of random graph orders have a “continuum limit.” We confirm that this is the case and show that the continuum limit is always a semiorder. Transitive percolation processes are a special case of a more general class called classical sequential growth models. We give a number of results describing the large‐scale structure of a general classical sequential growth model. We show that for any sufficiently large n, and any classical sequential growth model, there is a semiorder S on {0,…,n ‐ 1} such that the random partial order on {0,…,n ‐ 1} generated according to the model differs from S on an arbitrarily small proportion of pairs. We also show that, if any sequence of classical sequential growth models has a continuum limit, then this limit is (essentially) a semiorder. We give some examples of continuum limits that can occur. Classical sequential growth models were introduced as the only models satisfying certain properties making them suitable as discrete models for spacetime. Our results indicate that this class of models does not contain any that are good approximations to Minkowski space in any dimension ≥ 2. © 2009 Wiley Periodicals, Inc. Random Struct. Alg., 2010     

Coexistence of the infinite (*) clusters: — A remark on the square lattice site percolation

Summary We show that the critical probability p _c is strictly greater than 1/2 for the square lattice site percolation.     

Long range percolation in stationary point processes

We consider a continuum percolation model in ?d, d ? 1 in which any two points of a stationary point process are connected with a probability which decays exponentially in the distance between the points. We give sufficient conditions for the (non)-existence of a phase transition. We also give examples of processes which show that it is impossible to write down a theorem which relates the critical parameter value of a process to its density. Finally, we show that uniqueness of the infinite cluster is still valid in this general setting. © 1993 John Wiley & Sons, Inc.     

Bond percolation critical probability bounds for three Archimedean lattices

Rigorous bounds for the bond percolation critical probability are determined for three Archimedean lattices: .7385 < p_c((3, 12²) bond) < .7449, .6430 < p_c((4, 6, 12) bond) < .7376, .6281 < p_c((4, 8²) bond) < .7201. Consequently, the bond percolation critical probability of the (3, 12²) lattice is strictly larger than those of the other ten Archimedean lattices. Thus, the (3, 12²) bond percolation critical probability is possibly the largest of any vertex‐transitive graph with bond percolation critical probability that is strictly less than one. © 2002 Wiley Periodicals, Inc. Random Struct. Alg., 20: 507–518, 2002     

Exponential Decay for the Near-Critical Scaling Limit of the Planar Ising Model

We consider the Ising model at its critical temperature with external magnetic field ha^15/8 on the square lattice with lattice spacing a . We show that the truncated two-point function in this model decays exponentially with a rate independent of a as a ↓ 0 . As a consequence, we show exponential decay in the near-critical scaling limit Euclidean magnetization field. For the lattice model with a = 1 , the mass (inverse correlation length) is of order h^8/15 as h ↓ 0 ; for the Euclidean field, it equals exactly Ch^8/15 for some C . Although there has been much progress in the study of critical scaling limits, results on near-critical models are far fewer due to the lack of conformal invariance away from the critical point. Our arguments combine lattice and continuum FK representations, including coupled conformal loop and measure ensembles, showing that such ensembles can be useful even in the study of near-critical scaling limits. Thus we provide the first substantial application of measure ensembles. © 2020 The Authors. Communications on Pure and Applied Mathematics published by Wiley Periodicals LLC.     

Non-existence of phase transition of oriented percolation on Sierpinski carpet lattices

 A percolation problem on Sierpinski carpet lattices is considered. It is obtained that the critical probability of oriented percolation is equal to 1. In contrast it was already shown that the critical probability p _c of percolation is strictly less than 1 in Kumagai [9]. This result shows a difference between fractal-like lattice and ℤ ^d lattice. Received: 15 May 2002 / Revised version: 11 October 2002 / Published online: 21 February 2003 Mathematics Subject Classification (2000): Primary: 60K35, 82B43; Secondary: 82B26     

Carrier Method for the General Evaluation and Control of Pose, Molecular Conformation, Tracking, and the Like

The four basic geometric objects of points, point pairs, circles and spheres correspond to outer product null-spaces constructed by conformal points in conformal geometric algebra. Wedging with the infinity point we get four flat objects: finite-infinity point pairs, lines, planes and the whole 5D space. We show that all these basic geometric objects have the same algebraic structure. We then develop a single general algebraic method to quantify and interpolate the relative pose of the eight different classes of objects. As an explicit example we finally apply our framework to the conformation of organic macromolecules. Received: November, 2007. Accepted: February, 2008.     

On the critical percolation probabilities

Summary We prove that the critical probabilities of site percolation on the square lattice satisfy the relation p _c +p _c ^/* =1. Furthermore we prove the continuity of the function percolation probability.Work supported in part by U.S. National Science Foundation Grant PHY78-25390     

The first Steklov eigenvalue, conformal geometry, and minimal surfaces

We consider the relationship of the geometry of compact Riemannian manifolds with boundary to the first nonzero eigenvalue σ₁ of the Dirichlet-to-Neumann map (Steklov eigenvalue). For surfaces Σ with genus γ and k boundary components we obtain the upper bound σ₁L(∂Σ)?2(γ+k)π. For γ=0 and k=1 this result was obtained by Weinstock in 1954, and is sharp. We attempt to find the best constant in this inequality for annular surfaces (γ=0 and k=2). For rotationally symmetric metrics we show that the best constant is achieved by the induced metric on the portion of the catenoid centered at the origin which meets a sphere orthogonally and hence is a solution of the free boundary problem for the area functional in the ball. For a general class of (not necessarily rotationally symmetric) metrics on the annulus, which we call supercritical, we prove that σ₁(Σ)L(∂Σ) is dominated by that of the critical catenoid with equality if and only if the annulus is conformally equivalent to the critical catenoid by a conformal transformation which is an isometry on the boundary. Motivated by the annulus case, we show that a proper submanifold of the ball is immersed by Steklov eigenfunctions if and only if it is a free boundary solution. We then prove general upper bounds for conformal metrics on manifolds of any dimension which can be properly conformally immersed into the unit ball in terms of certain conformal volume quantities. We show that these bounds are only achieved when the manifold is minimally immersed by first Steklov eigenfunctions. We also use these ideas to show that any free boundary solution in two dimensions has area at least π, and we observe that this implies the sharp isoperimetric inequality for free boundary solutions in the two-dimensional case.     

Critical behavior in inhomogeneous random graphs

We study the critical behavior of inhomogeneous random graphs in the so‐called rank‐1 case, where edges are present independently but with unequal edge occupation probabilities. The edge occupation probabilities are moderated by vertex weights, and are such that the degree of vertex i is close in distribution to a Poisson random variable with parameter w_i, where w_i denotes the weight of vertex i. We choose the weights such that the weight of a uniformly chosen vertex converges in distribution to a limiting random variable W. In this case, the proportion of vertices with degree k is close to the probability that a Poisson random variable with random parameter W takes the value k. We pay special attention to the power‐law case, i.e., the case where \begin{align*}{\mathbb{P}}(W\geq k)\end{align*} is proportional to k^‐(τ‐1) for some power‐law exponent τ > 3, a property which is then inherited by the asymptotic degree distribution. We show that the critical behavior depends sensitively on the properties of the asymptotic degree distribution moderated by the asymptotic weight distribution W. Indeed, when \begin{align*}{\mathbb{P}}(W > k) \leq ck^{-(\tau-1)}\end{align*} for all k ≥ 1 and some τ > 4 and c > 0, the largest critical connected component in a graph of size n is of order n^2/3, as it is for the critical Erd?s‐Rényi random graph. When, instead, \begin{align*}{\mathbb{P}}(W > k)=ck^{-(\tau-1)}(1+o(1))\end{align*} for k large and some τ∈(3,4) and c > 0, the largest critical connected component is of the much smaller order n^{(τ‐2)/(τ‐1)}. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 42, 480–508, 2013     

Merging percolation on Zd and classical random graphs: Phase transition

We study a random graph model which is a superposition of bond percolation on Z^d with parameter p, and a classical random graph G(n,c/n). We show that this model, being a homogeneous random graph, has a natural relation to the so‐called “rank 1 case” of inhomogeneous random graphs. This allows us to use the newly developed theory of inhomogeneous random graphs to describe the phase diagram on the set of parameters c ≥ 0 and 0 ≤ p < p_c, where p_c = p_c(d) is the critical probability for the bond percolation on Z^d. The phase transition is of second order as in the classical random graph. We find the scaled size of the largest connected component in the supercritical regime. We also provide a sharp upper bound for the largest connected component in the subcritical regime. The latter is a new result for inhomogeneous random graphs with unbounded kernels. © 2009 Wiley Periodicals, Inc. Random Struct. Alg., 2010     

Platonic surfaces

If S _O is a Riemann surface with a complete metric of finite area and constant curvature -1, let S _C denote the conformal compactification of S _O. We show that, under the assumption that the cusps of S _O are large, there is a close relationship between the hyperbolic metrics on S _O and S _C. We use this relationship to show that , where the Platonic surface P _k is the conformal compactification of the modular surface S _k. Received: November, 1996; revised: February, 1998     

Aubin’s Lemma for the Yamabe constants of infinite coverings and a positive mass theorem

Aubin’s Lemma says that, if the Yamabe constant of a closed conformal manifold (M, C) is positive, then it is strictly less than the Yamabe constant of any of its non-trivial finite conformal coverings. We generalize this lemma to the one for the Yamabe constant of any (M _∞, C _∞) of its infinite conformal coverings, provided that π ₁(M) has a descending chain of finite index subgroups tending to π ₁(M _∞). Moreover, if the covering M _∞ is normal, the limit of the Yamabe constants of the finite conformal coverings (associated to the descending chain) is equal to that of (M _∞, C _∞). For the proof of this, we also establish a version of positive mass theorem for a specific class of asymptotically flat manifolds with singularities.     

The maximum of the conformal radius in the families of domains satifying additional conditions

We solve the problems on the maximum of the conformal radius R(D,1) in the family D(R₀) of all simply connected domains D ⊃ ℂ containing the points 0 and 1 and having a fixed value of the conformal radius R(D,0)=R₀, and in the family D(R₀, ρ) of domains from D(R₀) with given hyperbolic distance ρ=ρ_D(0,1) between 0 and 1. Analogs of the mentioned problems for doubly-connected domains with given conformal module are considered. Solution of the above problems is based on results of general character in the theory of problems of extremal decomposition and related module problems. Bibliography: 7 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 226, 1996, pp. 93–108.     

Phase Transition for the Speed of the Biased Random Walk on the Supercritical Percolation Cluster

We prove the sharpness of the phase transition for the speed in biased random walk on the supercritical percolation cluster on ?^d. That is, for each d ≥ 2, and for any supercritical parameter p > p_c, we prove the existence of a critical strength for the bias such that below this value the speed is positive, and above the value it is zero. We identify the value of the critical bias explicitly, and in the subballistic regime, we find the polynomial order of the distance moved by the particle. Each of these conclusions is obtained by investigating the geometry of the traps that are most effective at delaying the walk. A key element in proving our results is to understand that, on large scales, the particle trajectory is essentially one‐dimensional; we prove such a dynamic renormalization statement in a much stronger form than was previously known. © 2013 Wiley Periodicals, Inc.     

Non-linear Lie conformal algebras with three generators

We classify certain non-linear Lie conformal algebras with three generators, which can be viewed as deformations of the current Lie conformal algebra of sℓ ₂. In doing so we discover an interesting 1-parameter family of non-linear Lie conformal algebras and the corresponding freely generated vertex algebras , which includes for d = 1 the affine vertex algebra of sℓ ₂ at the critical level k = –2. We construct free-field realizations of the algebras extending the Wakimoto realization of at the critical level, and we compute their Zhu algebras. Dedicated to our teacher Victor Kac on the occasion of his 65th birthday