arXiv subjects

Peter Kleban

Publications and source records attributed to Peter Kleban.

At least 19 recordsLinked to original sources

arXiv2021

Multiple-SLE connectivity weights for rectangles, hexagons, and octagons

In a previous article, we define "connectivity weights" to be functions with these two properties: 1) They solve the three conformal Ward identities of conformal field theory (CFT) and a system of $2N$ null-state differential equations governing a CFT $2N$-point function of $ϕ_{1,2}$ or $ϕ_{2,1}$ primary Kac operators. 2) They satisfy a certain "duality" condition. In that same article, we argue that these functions are in fact pure partition functions for a multiple-SLE$_κ$ process with $2N$ curves, and we show how to find explicit formulas for them in terms of Coulomb gas contour integrals. However, this method gives very complicated formulas where simpler versions may be available, and it is not applicable for certain values of $κ\in(0,8)$ corresponding to well-known critical lattice models in statistical mechanics. In this article, we determine expressions for all connectivity weights for $N\in\{1,2,3,4\}$ (those with $N\in\{3,4\}$ are new) and for so-called "rainbow connectivity weights" for all $N\in\mathbb{Z}^++1$. We verify these formulas by explicitly showing that they satisfy the formal definition of a connectivity weight. In appendix B, we investigate logarithmic singularities of some of these expressions, appearing for certain values of $κ$ predicted by logarithmic CFT.

math-ph↗

arXiv2016

A formula for crossing probabilities of critical systems inside polygons

In this article, we generalize known formulas for crossing probabilities. Prior crossing results date back to J. Cardy's prediction of a formula for the probability that a percolation cluster in two dimensions connects the left and right sides of a rectangle at the percolation critical point in the continuum limit. Here, we predict a new formula for crossing probabilities of a continuum limit loop-gas model on a planar lattice inside a $2N$-sided polygon. In this model, boundary loops exit and then re-enter the polygon through its vertices, with exactly one loop passing once through each vertex, and these loops join the vertices pairwise in some specified connectivity through the polygon's exterior. The boundary loops also connect the vertices through the interior, which we regard as a crossing event. For particular values of the loop fugacity, this formula specializes to FK cluster (resp. spin cluster) crossing probabilities of a critical $Q$-state random cluster (resp. Potts) model on a lattice inside the polygon in the continuum limit. This includes critical percolation as the $Q=1$ random cluster model. These latter crossing probabilities are conditioned on a particular side-alternating free/fixed (resp. fluctuating/fixed) boundary condition on the polygon's perimeter, related to how the boundary loops join the polygon's vertices pairwise through the polygon's exterior in the associated loop-gas model. For $Q\in\{2,3,4\}$, we compare our predictions of these random cluster (resp. Potts) model crossing probabilities in a rectangle ($N=2$) and in a hexagon ($N=3$) with high-precision computer simulation measurements. We find that the measurements agree with our predictions very well for $Q\in\{2,3\}$ and reasonably well if $Q=4$.

cond-mat.stat-mech↗

arXiv2015

A solution space for a system of null-state partial differential equations 4

This article is the last of four that completely characterize a solution space $\mathcal{S}_N$ for a homogeneous system of $2N+3$ linear partial differential equations (PDEs) in $2N$ variables that arises in conformal field theory (CFT) and multiple Schramm-Loewner evolution (SLE). The system comprises $2N$ null-state equations and three conformal Ward identities that govern CFT correlation functions of $2N$ one-leg boundary operators. In the first two articles, we use methods of analysis and linear algebra to prove that $\dim\mathcal{S}_N\leq C_N$, with $C_N$ the $N$th Catalan number. Building on these results in the third article, we prove that $\dim\mathcal{S}_N=C_N$ and $\mathcal{S}_N$ is spanned by (real-valued) solutions constructed with the Coulomb gas (contour integral) formalism of CFT. In this article, we use these results to prove some facts concerning the solution space $\mathcal{S}_N$. First, we show that each of its elements equals a sum of at most two distinct Frobenius series in powers of the difference between two adjacent points (unless $8/κ$ is odd, in which case a logarithmic term may appear). This establishes an important element in the operator product expansion (OPE) for one-leg boundary operators, assumed in CFT. We also identify particular elements of $\mathcal{S}_N$, which we call connectivity weights, and exploit their special properties to conjecture a formula for the probability that the curves of a multiple-SLE$_κ$ process join in a particular connectivity. Finally, we propose a reason for why the "exceptional speeds" (certain $κ$ values that appeared in the analysis of the Coulomb gas solutions in the third article) and the minimal models of CFT are connected.

math-ph↗

arXiv2015

A solution space for a system of null-state partial differential equations 2

This article is the second of four that completely characterize a solution space $\mathcal{S}_N$ for a homogeneous system of $2N+3$ linear partial differential equations (PDEs) in 2N variables that arises in conformal field theory (CFT) and multiple Schramm-Lowner evolution (SLE). The system comprises $2N$ null-state equations and three conformal Ward identities which govern CFT correlation functions of $2N$ one-leg boundary operators. In the first article (part I), we use methods of analysis and linear algebra to prove that $\dim\mathcal{S}_N\leq C_N$, with $C_N$ the $N$th Catalan number. The analysis of that article is complete except for the proof of a lemma that it invokes. This article provides the proof. The lemma states that if every interval among $(x_2,x_3),$ $(x_3,x_4), ... ,(x_{2N-1},x_{2N})$ is a two-leg interval of $F\in\mathcal{S}_N$, then $F$ vanishes. Proving this lemma by contradiction, we show that the existence of such a nonzero function implies the existence of a non-vanishing CFT two-point function involving primary operators with different conformal weights, an impossibility. This proof (which is rigorous in spite of our occasional reference to CFT) involves two different types of estimates, those that give the asymptotic behavior of F as the length of one interval vanishes, and those that give this behavior as the lengths of two intervals vanish simultaneously. We derive these estimates by using Green functions to rewrite certain null-state PDEs as integral equations, combining other null-state PDEs to obtain Schauder interior estimates, and then repeatedly integrating the integral equations with these estimates until we obtain optimal bounds. In the case where two adjacent interval lengths vanish, we use a Green function that contains the Jacobi heat kernel as its essential ingredient.

math-ph↗

arXiv2015

A solution space for a system of null-state partial differential equations 1

In this first of four articles, we study a homogeneous system of $2N+3$ linear partial differential equations (PDEs) in $2N$ variables that arises in conformal field theory (CFT) and multiple Schramm-Lowner evolution (SLE). In CFT, these are null-state equations and conformal Ward identities. They govern partition functions for the continuum limit of a statistical cluster or loop model, such as percolation, or more generally the Potts models and O$(n)$ models, at the statistical mechanical critical point. (SLE partition functions also satisfy these equations.) For such a lattice model in a polygon $\mathcal{P}$ with its $2N$ sides exhibiting a free/fixed side-alternating boundary condition, this partition function is proportional to the CFT correlation function $\langleψ_1^c(w_1)ψ_1^c(w_2)\dotsmψ_1^c(w_{2N-1})ψ_1^c(w_{2N})\rangle^{\mathcal{P}}$ where the $w_i$ are the vertices of $\mathcal{P}$ and where $ψ_1^c$ is a one-leg corner operator. When conformally mapped onto the upper half-plane, methods of CFT show that this correlation function satisfies the system of PDEs that we consider. This article is the first of four that completely and rigorously characterize the space of all solutions for this system of PDEs that grow no faster than a power law. In this first article, we use methods of analysis to prove that the dimension of this solution space is no more than $C_N$, the $N$th Catalan number. This proof is contained entirely within this article, except for the proof of lemma 14, which constitutes the second article ("part II"). In the third article ("part III"), we use the results of this article to prove that the solution space of this system of PDEs has dimension $C_N$ and is spanned by solutions constructed with the CFT Coulomb gas (contour integral) formalism. In the fourth article ("part IV"), we prove further CFT-related properties about these solutions.

math-ph↗

arXiv2015

A solution space for a system of null-state partial differential equations 3

This article is the third of four that completely characterize a solution space $\mathcal{S}_N$ for a homogeneous system of $2N+3$ linear partial differential equations (PDEs) in $2N$ variables that arises in conformal field theory (CFT) and multiple Schramm-Lowner evolution (SLE). The system comprises $2N$ null-state equations and three conformal Ward identities that govern CFT correlation functions of $2N$ one-leg boundary operators. In the previous two articles (parts I and II), we use methods of analysis and linear algebra to prove that $\dim\mathcal{S}_N\leq C_N$, with $C_N$ the $N$th Catalan number. Extending these results, we prove in this article that $\dim\mathcal{S}_N=C_N$ and $\mathcal{S}_N$ entirely consists of (real-valued) solutions constructed with the CFT Coulomb gas (contour integral) formalism. In order to prove this claim, we show that a certain set of $C_N$ such solutions is linearly independent. Because the formulas for these solutions are complicated, we prove linear independence indirectly. We use the linear injective map of lemma 15 in part I to send each solution of the mentioned set to a vector in $\mathbb{R}^{C_N}$, whose components we find as inner products of elements in a Temperley-Lieb algebra. We gather these vectors together as columns of a symmetric $C_N$ by $C_N$ matrix, with the form of a meander matrix. If the determinant of this matrix does not vanish, then the set of $C_N$ Coulomb gas solutions is linearly independent. And if this determinant does vanish, then we construct an alternative set of $C_N$ Coulomb gas solutions and follow a similar procedure to show that this set is linearly independent. The latter situation is closely related to CFT minimal models.

math-ph↗

arXiv2012

Cluster pinch-point densities in polygons

In a statistical cluster or loop model such as percolation, or more generally the Potts models or O(n) models, a pinch point is a single bulk point where several distinct clusters or loops touch. In a polygon P harboring such a model in its interior and with 2N sides exhibiting free/fixed side-alternating boundary conditions, "boundary" clusters anchor to the fixed sides of P. At the critical point and in the continuum limit, the density (i.e., frequency of occurrence) of pinch-point events between s distinct boundary clusters at a bulk point w in P is proportional to _P. The w_i are the vertices of P, psi_1^c is a conformal field theory (CFT) corner one-leg operator, and Psi_s is a CFT bulk 2s-leg operator. In this article, we use the Coulomb gas formalism to construct explicit contour integral formulas for these correlation functions and thereby calculate the density of various pinch-point configurations at arbitrary points in the rectangle, in the hexagon, and for the case s=N, in the 2N-sided polygon at the system's critical point. Explicit formulas for these results are given in terms of algebraic functions or integrals of algebraic functions, particularly Lauricella functions. In critical percolation, the result for s=N=2 gives the density of red bonds between boundary clusters (in the continuum limit) inside a rectangle. We compare our results with high-precision simulations of critical percolation and Ising FK clusters in a rectangle of aspect ratio two and in a regular hexagon and find very good agreement.

math-ph↗

arXiv2011

Complete conformal field theory solution of a chiral six-point correlation function

Using conformal field theory, we perform a complete analysis of the chiral six-point correlation function C(z)=< ϕ_{1,2}ϕ_{1,2} Φ_{1/2,0}(z, \bar z) ϕ_{1,2}ϕ_{1,2} >, with the four ϕ_{1,2} operators at the corners of an arbitrary rectangle, and the point z = x+iy in the interior. We calculate this for arbitrary central charge (equivalently, SLE parameter κ> 0). C is of physical interest because percolation (κ= 6) and many other two-dimensional critical points, it specifies the density at z of critical clusters conditioned to touch either or both vertical ends of the rectangle, with these ends `wired', i.e. constrained to be in a single cluster, and the horizontal ends free. The correlation function may be written as the product of an algebraic prefactor f and a conformal block G, where f = f(x,y,m), with m a cross-ratio specified by the corners (m determines the aspect ratio of the rectangle). By appropriate choice of f and using coordinates that respect the symmetry of the problem, the conformal block G is found to be independent of either y or x, and given by an Appell function.

cond-mat.stat-mech↗

arXiv2011

Cluster densities at 2-D critical points in rectangular geometries

Making use of a recent complete calculation of a chiral six-point correlation function C(z) in a rectangle we calculate various quantities of interest for percolation (SLE parameter κ= 6) and many other two-dimensional critical points. In particular, we specify the density at z of critical clusters conditioned to touch either or both vertical sides of the rectangle, with these sides 'wired,' i.e. constrained to be in a single cluster, and the horizontal sides free. These quantities probe the structure of various cluster configurations, including those that contribute to the crossing probability. We first examine the effects of boundary conditions on C for the critical O(n) loop models in both high and low density phases and for both Fortuin-Kasteleyn (FK) and spin clusters in the critical Q-state Potts models. A Coulomb gas analysis then allows us to calculate the cluster densities with various conditionings in terms of the known conformal blocks. Explicit formulas generalizing Cardy's horizontal crossing probability to these models (using previously known results) are also presented. These solutions are employed to generalize previous results demonstrating factorization of higher-order correlation functions to the critical systems mentioned. An explicit formula for the density of critical percolation clusters that cross a rectangle horizontally with free boundary conditions is also given. Simplifications of the hypergeometric functions in our solutions for various models are presented. High precision simulations verify these predictions for percolation and for the Q=2 and 3-state Potts models, including both FK and spin clusters. Our formula for the density of crossing clusters in percolation in open systems is also verified.

math-ph↗

arXiv2011

Factorization of correlations in two-dimensional percolation on the plane and torus

Recently, Delfino and Viti have examined the factorization of the three-point density correlation function P_3 at the percolation point in terms of the two-point density correlation functions P_2. According to conformal invariance, this factorization is exact on the infinite plane, such that the ratio R(z_1, z_2, z_3) = P_3(z_1, z_2, z_3) [P_2(z_1, z_2) P_2(z_1, z_3) P_2(z_2, z_3)]^{1/2} is not only universal but also a constant, independent of the z_i, and in fact an operator product expansion (OPE) coefficient. Delfino and Viti analytically calculate its value (1.022013...) for percolation, in agreement with the numerical value 1.022 found previously in a study of R on the conformally equivalent cylinder. In this paper we confirm the factorization on the plane numerically using periodic lattices (tori) of very large size, which locally approximate a plane. We also investigate the general behavior of R on the torus, and find a minimum value of R approx. 1.0132 when the three points are maximally separated. In addition, we present a simplified expression for R on the plane as a function of the SLE parameter kappa.

cond-mat.dis-nn↗

arXiv2009

Intervals Between Farey Fractions in the Limit of Infinite Level

The modified Farey sequence consists, at each level k, of rational fractions r_k^{(n)}, with n=1, 2, ...,2^k+1. We consider I_k^{(e)}, the total length of (one set of) alternate intervals between Farey fractions that are new (i.e., appear for the first time) at level k, I^{(e)}_k := \sum_{i=1}^{2^{k-2}} (r_k^{(4i)}- r_k^{(4i-2)}) . We prove that \liminf_{k\to \infty} I_k^{(e)}=0, and conjecture that in fact \lim_{k \to \infty}I_k^{(e)}=0. This simple geometrical property of the Farey fractions turns out to be surprisingly subtle, with no apparent simple interpretation. The conjecture is equivalent to $ lim_{k \to \infty}S_{k}=0, where S_{k} is the sum over the inverse squares of the new denominators at level k, S_{k}:=\sum_{n=1}^{2^{k-1}} 1/ (d_k^{(2n)} )^2. Our result makes use of bounds for Farey fraction intervals in terms of their "parent" intervals at lower levels.

math-ph↗

arXiv2009

General solution of an exact correlation function factorization in conformal field theory

We discuss a correlation function factorization, which relates a three-point function to the square root of three two-point functions. This factorization is known to hold for certain scaling operators at the two-dimensional percolation point and in a few other cases. The correlation functions are evaluated in the upper half-plane (or any conformally equivalent region) with operators at two arbitrary points on the real axis, and a third arbitrary point on either the real axis or in the interior. This type of result is of interest because it is both exact and universal, relates higher-order correlation functions to lower-order ones, and has a simple interpretation in terms of cluster or loop probabilities in several statistical models. This motivated us to use the techniques of conformal field theory to determine the general conditions for its validity. Here, we discover a correlation function which factorizes in this way for any central charge c, generalizing previous results. In particular, the factorization holds for either FK (Fortuin-Kasteleyn) or spin clusters in the Q-state Potts models; it also applies to either the dense or dilute phases of the O(n) loop models. Further, only one other non-trivial set of highest-weight operators (in an irreducible Verma module) factorizes in this way. In this case the operators have negative dimension (for c < 1) and do not seem to have a physical realization.

cond-mat.stat-mech↗

arXiv2009

New percolation crossing formulas and second-order modular forms

We consider the three new crossing probabilities for percolation recently found via conformal field theory by Simmons, Kleban and Ziff. We prove that all three of them (i) may be simply expressed in terms of Cardy's and Watts' crossing probabilities, (ii) are (weakly holomorphic) second-order modular forms of weight 0 (and a single particular type) on the congruence group $Γ(2)$, and (iii) under some technical assumptions (similar to those used by Kleban and Zagier, are completely determined by their transformation laws. The only physical input in (iii) is Cardy's crossing formula, which suggests an unknown connection between all crossing-type formulas.

math-ph↗

arXiv2008

Factorization of percolation density correlation functions for clusters touching the sides of a rectangle

We consider the density at a point z = x + i y of critical percolation clusters that touch the left [P_L(z)], right [P_R(z)], or both [P_{LR}(z)] sides of a rectangular system, with open boundary conditions on the top and bottom. The ratio C(z) = P_{LR}(z) / sqrt[P_L(z) P_R(z) Pi_h], where Pi_h is the probability of horizontal crossing given by Cardy, is a universal function of z and goes to a constant value C_0 = 2^(7/2) 3^(-3/4) pi^(5/2) Gamma(1/3)^(-9/2) = 1.0299268... far from the ends. We observe numerically that C(z) depends upon x but not y for wired b.c., and this result leads to an explicit expression for C(z) via conformal field theory. For the semi-infinite strip we also derive explict expressions for P_L(z), P_R(z), and P_{LR}(z), for both wired and open b.c. Our results enable calculation of the finite-size corrections to the factorization near an isolated anchor point, for the case of clusters anchored at two boundary points. Finally, we present numerical results for a rectangle with periodic b.c. in the horizontal direction, and find that C(z) approaches a constant value C_1 = 1.022.

cond-mat.stat-mech↗

arXiv2008

First Column Boundary Operator Product Expansion Coefficients

We calculate boundary operator product expansion coefficients for boundary operators in the first column of the Kac table in conformal field theories. For c=0 we give closed form expressions for all such coefficients. Then we generalize to the augmented minimal models, giving explicit expressions for coefficients valid when ϕ_{1,2} mediates a change from fixed to free boundary conditions. These quantities are determined by computing an arbitrary four-point correlation function of first column operators. Our calculation first determines the appropriate (non-logarithmic) conformal blocks by using standard null-vector methods. The behavior of these blocks under crossing symmetry then provides a general closed form expression for the desired coefficients, as a product of ratios of gamma functions. This calculation was inspired by the need for several of these coefficients in certain correlation function formulas for critical two-dimensional percolation and the augmented q=2 and q=3 state critical Potts models.

cond-mat.stat-mech↗

arXiv2007

Exact factorization of correlation functions in 2-D critical percolation

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.

math-ph↗

arXiv2007

Percolation Crossing Formulas and Conformal Field Theory

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 support this idea and suggest the presence of additional symmetry in $c=0$ conformal field theories.

cond-mat.stat-mech↗

arXiv2007

The density of critical percolation clusters touching the boundaries of strips and squares

We consider the density of two-dimensional critical percolation clusters, constrained to touch one or both boundaries, in infinite strips, half-infinite strips, and squares, as well as several related quantities for the infinite strip. Our theoretical results follow from conformal field theory, and are compared with high-precision numerical simulation. For example, we show that the density of clusters touching both boundaries of an infinite strip of unit width (i.e. crossing clusters) is proportional to $(\sin πy)^{-5/48}\{[\cos(πy/2)]^{1/3} +[\sin (πy/2)]^{1/3}-1\}$. We also determine numerically contours for the density of clusters crossing squares and long rectangles with open boundaries on the sides, and compare with theory for the density along an edge.

cond-mat.dis-nn↗