Continuum Scaling Limit Research Articles

The multiplicative coalescent, inhomogeneous continuum random trees, and new universality classes for critical random graphs

One major open conjecture in the area of critical random graphs, formulated by statistical physicists, and supported by a large amount of numerical evidence over the last decade [23, 24, 28, 63] is as follows: for a wide array of random graph models with degree exponent $\tau\in (3,4)$, distances between typical points both within maximal components in the critical regime as well as on the minimal spanning tree on the giant component in the supercritical regime scale like $n^{(\tau-3)/(\tau-1)}$. In this paper we study the metric space structure of maximal components of the multiplicative coalescent, in the regime where the sizes converge to excursions of L\'evy processes without replacement [10], yielding a completely new class of limiting random metric spaces. A by-product of the analysis yields the continuum scaling limit of one fundamental class of random graph models with degree exponent $\tau\in (3,4)$ where edges are rescaled by $n^{-(\tau-3)/(\tau-1)}$ yielding the first rigorous proof of the above conjecture. The limits in this case are compact tree-like random fractals with finite fractal dimensions and with a dense collection of hubs (infinite degree vertices) a finite number of which are identified with leaves to form shortcuts. It is generally believed that dynamic versions of a number of fundamental random graph models, as one moves from the barely subcritical to the critical regime can be approximated by the multiplicative coalescent. In work in progress, the general theory developed in this paper is used to prove analogous limit results for other random graph models with degree exponent $\tau\in (3,4)$. Our proof makes crucial use of inhomogeneous continuum random trees (ICRT), which have previously arisen in the study of the entrance boundary of the additive coalescent. We show that tilted versions of the same objects using the associated mass measure, describe connectivity properties of the multiplicative coalescent. Since convergence of height processes of corresponding approximating p-trees is not known, we use general methodology in [14] and develop novel techniques relying on first showing convergence in the Gromov-weak topology and then extending this to Gromov-Hausdorff-Prokhorov convergence by proving a global lower mass-bound. Keywords: Multiplicative coalescent, p-trees, inhomogeneous continuum random trees, critical random graphs, Gromov-Hausdorff distance, Gromov-weak topology

Critical dense polymers with Robin boundary conditions, half-integer Kac labels and [formula omitted] fermions

For general Temperley–Lieb loop models, including the logarithmic minimal models LM(p,p′) with p,p′ coprime integers, we construct an infinite family of Robin boundary conditions on the strip as linear combinations of Neumann and Dirichlet boundary conditions. These boundary conditions are Yang–Baxter integrable and allow loop segments to terminate on the boundary. Algebraically, the Robin boundary conditions are described by the one-boundary Temperley–Lieb algebra. Solvable critical dense polymers is the first member LM(1,2) of the family of logarithmic minimal models and has loop fugacity β=0 and central charge c=−2. Specialising to LM(1,2) with our Robin boundary conditions, we solve the model exactly on strips of arbitrary finite size N and extract the finite-size conformal corrections using an Euler–Maclaurin formula. The key to the solution is an inversion identity satisfied by the commuting double row transfer matrices. This inversion identity is established directly in the Temperley–Lieb algebra. We classify the eigenvalues of the double row transfer matrices using the physical combinatorics of the patterns of zeros in the complex spectral parameter plane and obtain finitised characters related to spaces of coinvariants of Z4 fermions. In the continuum scaling limit, the Robin boundary conditions are associated with irreducible Virasoro Verma modules with conformal weights Δr,s−12=132(L2−4) where L=2s−1−4r, r∈Z, s∈N. These conformal weights populate a Kac table with half-integer Kac labels. Fusion of the corresponding modules with the generators of the Kac fusion algebra is examined and general fusion rules are proposed.

Logarithmic superconformal minimal models

The higher fusion level logarithmic minimal models have recently been constructed as the diagonal GKO cosets where n ≥ 1 is an integer fusion level and k = nP/(P′− P) − 2 is a fractional level. For n = 1, these are the well-studied logarithmic minimal models . For n ≥ 2, we argue that these critical theories are realized on the lattice by n × n fusion of the n = 1 models. We study the critical fused lattice models within a lattice approach and focus our study on the n = 2 models. We call these logarithmic superconformal minimal models where P = |2p − p′|, P′ = p′ and p, p′ are coprime. These models share the central charges of the rational superconformal minimal models . Lattice realizations of these theories are constructed by fusing 2 × 2 blocks of the elementary face operators of the n = 1 logarithmic minimal models . Algebraically, this entails the fused planar Temperley–Lieb algebra which is a spin-1 Birman–Murakami–Wenzl tangle algebra with loop fugacity β2 = [x]3 = x2 + 1 + x−2 and twist ω = x4 where x = eiλ and λ = (p′− p)π/p′. The first two members of this n = 2 series are superconformal dense polymers with , β2 = 0 and superconformal percolation with c = 0, β2 = 1. We calculate the bulk and boundary free energies analytically. By numerically studying finite-size conformal spectra on the strip with appropriate boundary conditions, we argue that, in the continuum scaling limit, these lattice models are associated with the logarithmic superconformal models . For system size N, we propose finitized Kac character formulae of the form for s-type boundary conditions with r = 1, s = 1,2,3, …, ℓ = 0,1,2. The P, P′ dependence enters only in the fractional power of q in the prefactor and ℓ = 0,2 label the Neveu–Schwarz sectors (r + s even) and ℓ = 1 labels the Ramond sectors (r + s odd). Combinatorially, the finitized characters involve Motzkin and Riordan polynomials defined in terms of q-trinomial coefficients. Using the Hamiltonian limit and the finitized characters, we argue, from examples of finite-lattice calculations, that there exist reducible yet indecomposable representations for which the Virasoro dilatation operator L0 exhibits rank-2 Jordan cells, confirming that these theories are indeed logarithmic. We relate these results to the N = 1 superconformal representation theory.

Modular invariant partition function of critical dense polymers

A lattice model of critical dense polymers is solved exactly for arbitrary system size on the torus. More generally, an infinite family of lattice loop models is studied on the torus and related to the corresponding Fortuin–Kasteleyn random cluster models. Starting with a cylinder, the commuting periodic single-row transfer matrices are built from the periodic Temperley–Lieb algebra extended by the shift operators Ω±1. In this enlarged algebra, the non-contractible loop fugacity is α and the contractible loop fugacity is β. The torus is formed by gluing the top and bottom of the cylinder. This gives rise to a variety of non-contractible loops winding around the torus. Because of their nonlocal nature, the standard matrix trace does not produce the proper geometric torus. Instead, we introduce a modified matrix trace for this purpose. This is achieved by using a representation of the enlarged periodic Temperley–Lieb algebra with a parameter v that keeps track of the winding of defects on the cylinder. The transfer matrix representatives and their eigenvalues thus depend on v. The modified trace is constructed as a linear functional on planar connectivity diagrams in terms of matrix traces Trd (with a fixed number of defects d) and Chebyshev polynomials of the first kind. For critical dense polymers, where β=0, the transfer matrix eigenvalues are obtained by solving a functional equation in the form of an inversion identity. The solution depends on d and is subject to selection rules which we prove. Simplifications occur if all non-contractible loop fugacities are set to α=2 in which case the traces are evaluated at v=1. In the continuum scaling limit, the corresponding conformal torus partition function obtained from finite-size corrections agrees with the known modular invariant partition function of symplectic fermions.

Refined conformal spectra in the dimer model

Working with Lieb’s transfer matrix for the dimer model, we point out that the full set of dimer configurations may be partitioned into disjoint subsets (sectors) closed under the action of the transfer matrix. These sectors are labelled by an integer or half-integer quantum number we call the variation index. In the continuum scaling limit, each sector gives rise to a representation of the Virasoro algebra. We determine the corresponding conformal partition functions and their finitizations, and observe an intriguing link to the Ramond and Neveu–Schwarz sectors of the critical dense polymer model as described by a conformal field theory with central charge c =− 2.

Off-critical logarithmic minimal models

We consider the integrable minimal models ℳ(m,m′;t), corresponding to the φ1,3 perturbation off-criticality, in the logarithmic limit m,m′ → ∞, m/m′ → p/p′ where p,p′ are coprime and the limit is taken through coprime values of m,m′. We view these off-critical minimal models ℳ(m,m′;t) as the continuum scaling limit of the Forrester–Baxter restricted solid-on-solid (RSOS) models on the square lattice. Applying corner transfer matrices to the Forrester–Baxter RSOS models in regime III, we argue that taking first the thermodynamic limit and second the logarithmic limit yields off-critical logarithmic minimal models ℒℳ(p,p′;t) corresponding to the φ1,3 perturbation of the critical logarithmic minimal models ℒℳ(p,p′). Specifically, in accord with the Kyoto correspondence principle, we show that the logarithmic limit of the one-dimensional configurational sums yields finitized quasi-rational characters of the Kac representations of the critical logarithmic minimal models ℒℳ(p,p′). We also calculate the logarithmic limit of certain off-critical observables \U0001d4aar,s related to one-point functions and show that the associated critical exponents produce all conformal dimensions in the infinitely extended Kac table. The corresponding Kac labels (r,s) satisfy (ps−p′r)2 < 8p(p′ − p). The exponent 2 − α = p′/2(p′ − p) is obtained from the logarithmic limit of the free energy giving the conformal dimension for the perturbing field t. As befits a non-unitary theory, some observables \U0001d4aar,s diverge at criticality.

Triviality problem and high-temperature expansions of higher susceptibilities for the Ising and scalar-field models in four-, five-, and six-dimensional lattices

High-temperature expansions are presently the only viable approach to the numerical calculation of the higher susceptibilities for the spin and the scalar-field models on high-dimensional lattices. The critical amplitudes of these quantities enter into a sequence of universal amplitude ratios that determine the critical equation of state. We have obtained a substantial extension, through order 24, of the high-temperature expansions of the free energy (in presence of a magnetic field) for the Ising models with spin s≥1/2 and for the lattice scalar-field theory with quartic self-interaction on the simple-cubic and the body-centered-cubic lattices in four, five, and six spatial dimensions. A numerical analysis of the higher susceptibilities obtained from these expansions yields results consistent with the widely accepted ideas, based on the renormalization group and the constructive approach to Euclidean quantum field theory, concerning the no-interaction ("triviality") property of the continuum (scaling) limit of spin-s Ising and lattice scalar-field models at and above the upper critical dimensionality.

THE KARDAR–PARISI–ZHANG EQUATION AND UNIVERSALITY CLASS

Brownian motion is a continuum scaling limit for a wide class of random processes, and there has been great success in developing a theory for its properties (such as distribution functions or regularity) and expanding the breadth of its universality class. Over the past 25 years a new universality class has emerged to describe a host of important physical and probabilistic models (including one-dimensional interface growth processes, interacting particle systems and polymers in random environments) which display characteristic, though unusual, scalings and new statistics. This class is called the Kardar–Parisi–Zhang (KPZ) universality class and underlying it is, again, a continuum object — a non-linear stochastic partial differential equation — known as the KPZ equation.The purpose of this survey is to explain the context for, as well as the content of a number of mathematical breakthroughs which have culminated in the derivation of the exact formula for the distribution function of the KPZ equation started with narrow wedge initial data. In particular we emphasize three topics: (1) The approximation of the KPZ equation through the weakly asymmetric simple exclusion process; (2) The derivation of the exact one-point distribution of the solution to the KPZ equation with narrow wedge initial data; (3) Connections with directed polymers in random media.As the purpose of this article is to survey and review, we make precise statements but provide only heuristic arguments with indications of the technical complexities necessary to make such arguments mathematically rigorous.

-extended Kac representations and integrable boundary conditions in the logarithmic minimal models

We construct new Yang–Baxter integrable boundary conditions in the lattice approach to the logarithmic minimal model giving rise to reducible yet indecomposable representations of rank 1 in the continuum scaling limit. We interpret these -extended Kac representations as finitely generated -extended Feigin–Fuchs modules over the triplet W-algebra . The -extended fusion rules of these representations are inferred from the recently conjectured Virasoro fusion rules of the Kac representations in the underlying logarithmic minimal model . We also introduce the modules contragredient to the -extended Kac modules and work out the correspondingly extended fusion algebra. Our results are in accordance with the Kazhdan–Lusztig dual of tensor products of modules over the restricted quantum universal enveloping algebra at q = eπi/p. Finally, polynomial fusion rings isomorphic with the various fusion algebras are determined, and the corresponding Grothendieck ring of characters is identified.

Physical combinatorics and quasiparticles

We consider the physical combinatorics of critical lattice models and their associatedconformal field theories arising in the continuum scaling limit. As examples, we considerA-type unitary minimal models and the level-1 Wess–Zumino–Witten (WZW) model. The Hamiltonian of the WZW model is the invariant XXX quantum spin chain. For simplicity, we consider these theories only in theirvacuum sectors on the strip. Combinatorially, fermionic particles are introduced ascertain features of RSOS paths. They are composites of dual-particles and exhibitthe properties of quasiparticles. The particles and dual-particles are identified,through a conjectured energy-preserving bijection, with patterns of zeros of theeigenvalues of the fused transfer matrices in their analyticity strips. The associated(m,n) systems arise as geometric packing constraints on the particles. The analyticity encodedin the patterns of zeros is the key to the analytic calculation of the excitationenergies through the thermodynamic Bethe ansatz (TBA). As a by-product ofour study, in the case of the WZW or XXX model, we find a relation betweenthe location of the Bethe root strings and the location of the transfer matrix2-strings.

Formula omitted]-extended logarithmic minimal models

We consider the continuum scaling limit of the infinite series of Yang–Baxter integrable logarithmic minimal models LM ( p , p ′ ) as ‘rational’ logarithmic conformal field theories with extended W symmetry. The representation content is found to consist of 6 p p ′ − 2 p − 2 p ′ W -indecomposable representations of which 2 p + 2 p ′ − 2 are of rank 1, 4 p p ′ − 2 p − 2 p ′ are of rank 2, while the remaining 2 ( p − 1 ) ( p ′ − 1 ) are of rank 3. We identify these representations with suitable limits of Yang–Baxter integrable boundary conditions on the lattice. The W -indecomposable rank-1 representations are all W -irreducible while we present a conjecture for the embedding patterns of the W -indecomposable rank-2 and -3 representations. The associated W -extended characters are all given explicitly and decompose as finite non-negative sums of W -irreducible characters. The latter correspond to W -irreducible subfactors and we find that there are 2 p p ′ + ( p − 1 ) ( p ′ − 1 ) / 2 of them. We present fermionic character expressions for some of the rank-2 and all of the rank-3 W -indecomposable representations. To distinguish between inequivalent W -indecomposable representations of identical characters, we introduce ‘refined’ characters carrying information also about the Jordan-cell content of a representation. Using a lattice implementation of fusion on a strip, we study the fusion rules for the W -indecomposable representations and find that they generate a closed fusion algebra, albeit one without identity for p > 1 . We present the complete set of fusion rules and interpret the closure of this fusion algebra as confirmation of the proposed extended symmetry. Finally, 2 p p ′ of the W -indecomposable representations are in fact W -projective representations and they generate a closed fusion subalgebra.

-extended fusion algebra of critical percolation

Two-dimensional critical percolation is the member of the infinite series of Yang–Baxter integrable logarithmic minimal models . We consider the continuum scaling limit of this lattice model as a ‘rational’ logarithmic conformal field theory with extended symmetry and use a lattice approach on a strip to study the fundamental fusion rules in this extended picture. We find that the representation content of the ensuing closed fusion algebra contains 26 -indecomposable representations with eight rank-1 representations, 14 rank-2 representations and four rank-3 representations. We identify these representations with suitable limits of Yang–Baxter integrable boundary conditions on the lattice and obtain their associated -extended characters. The latter decompose as finite non-negative sums of -irreducible characters of which 13 are required. Implementation of fusion on the lattice allows us to read off the fusion rules governing the fusion algebra of the 26 representations and to construct an explicit Cayley table. The closure of these representations among themselves under fusion is remarkable confirmation of the proposed extended symmetry.

Two-Dimensional Critical Percolation: The Full Scaling Limit

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.

Scaling Limit and Critical Exponents for Two-Dimensional Bootstrap Percolation

Consider a cellular automaton with state space {0,1} ℤ2 where the initial configuration ω_0 is chosen according to a Bernoulli product measure, 1’s are stable, and 0’s become 1’s if they are surrounded by at least three neighboring 1’s. In this paper we show that the configuration ω_n at time n converges exponentially fast to a final configuration $$\bar\omega$$ , and that the limiting measure corresponding to $$\bar\omega$$ is in the universality class of Bernoulli (independent) percolation. More precisely, assuming the existence of the critical exponents β, η, ν and γ, and of the continuum scaling limit of crossing probabilities for independent site percolation on the close-packed version of ℤ2 (i.e. for independent *-percolation on ℤ), we prove that the bootstrapped percolation model has the same scaling limit and critical exponents.This type of bootstrap percolation can be seen as a paradigm for a class of cellular automata whose evolution is given, at each time step, by a monotonic and nonessential enhancement [Aizenman and Grimmett, J. Stat. Phys. 63: 817--835 (1991); Grimmett, Percolation, 2nd Ed. (Springer, Berlin, 1999)

EXCITED BOUNDARY TBA IN THE TRICRITICAL ISING MODEL

By considering the continuum scaling limit of the A4 RSOS lattice model of Andrews-Baxter-Forrester with integrable boundaries, we derive excited state TBA equations describing the boundary flows of the tricritical Ising model. Fixing the bulk weights to their critical values, the integrable boundary weights admit two parameters ξ and η which play the role of the perturbing boundary fields φ1,3 and φ1,2 and induce the renormalization group flow between boundary fixed points. The boundary TBA equations determining the RG flows are derived in the [Formula: see text] example. The induced map between distinct Virasoro characters of the theory are specified in terms of distribution of zeros of the double row transfer matrix.

Random trees between two walls: exact partition function

We derive the exact partition function for a discrete model of random trees embedded in a one-dimensional space. These trees have vertices labeled by integers representing their position in the target space, with the SOS constraint that adjacent vertices have labels differing by +1 or -1. A non-trivial partition function is obtained whenever the target space is bounded by walls. We concentrate on the two cases where the target space is (i) the half-line bounded by a wall at the origin or (ii) a segment bounded by two walls at a finite distance. The general solution has a soliton-like structure involving elliptic functions. We derive the corresponding continuum scaling limit which takes the remarkable form of the Weierstrass p-function with constrained periods. These results are used to analyze the probability for an evolving population spreading in one dimension to attain the boundary of a given domain with the geometry of the target (i) or (ii). They also translate, via suitable bijections, into generating functions for bounded planar graphs.

Exact ϕ1,3 boundary flows in the tricritical Ising model

We consider the tricritical Ising model on a strip or cylinder under the integrable perturbation by the thermal ϕ 1,3 boundary field. This perturbation induces five distinct renormalization group (RG) flows between Cardy type boundary conditions labelled by the Kac labels ( r, s). We study these boundary RG flows in detail for all excitations. Exact thermodynamic Bethe ansatz (TBA) equations are derived using the lattice approach by considering the continuum scaling limit of the A 4 lattice model with integrable boundary conditions. Fixing the bulk weights to their critical values, the integrable boundary weights admit a thermodynamic boundary field ξ which induces the flow and, in the continuum scaling limit, plays the role of the perturbing boundary field ϕ 1,3. The excitations are completely classified, in terms of string content, by ( m, n) systems and quantum numbers but the string content changes by either two or three well-defined mechanisms along the flow. We identify these mechanisms and obtain the induced maps between the relevant finitized Virasoro characters. We also solve the TBA equations numerically to determine the boundary flows for the leading excitations.

Excited TBA equations II: massless flow from tricritical to critical Ising model

We consider the massless tricritical Ising model M(4,5) perturbed by the thermal operator ϕ 1,3 in a cylindrical geometry and apply integrable boundary conditions, labelled by the Kac labels ( r, s), that are natural off-critical perturbations of known conformal boundary conditions. We derive massless thermodynamic Bethe ansatz (TBA) equations for all excitations by solving, in the continuum scaling limit, the TBA functional equation satisfied by the double-row transfer matrices of the A 4 lattice model of Andrews, Baxter and Forrester (ABF) in Regime IV. The resulting TBA equations describe the massless renormalization group flow from the tricritical to critical Ising model. As in the massive case of Part I, the excitations are completely classified in terms of ( m , n ) systems but the string content changes by one of three mechanisms along the flow. Using generalized q-Vandermonde identities, we show that this leads to a flow from tricritical to critical Ising characters. The excited TBA equations are solved numerically to follow the continuous flows from the UV to the IR conformal fixed points.

Integrable and conformal twisted boundary conditions forsl(2)A-D-Elattice models

We study integrable realizations of conformal twisted boundary conditions for sl(2) unitary minimal models on a torus. These conformal field theories are realized as the continuum scaling limit of critical G = A,D,E lattice models with positive spectral parameter u > 0 and Coxeter number g. Integrable seams are constructed by fusing blocks of elementary local face weights. The usual A-type fusions are labelled by the Kac labels (r, s) and are associated with the Verlinde fusion algebra. We introduce a new type of fusion in the two braid limits u->\pm i\infty associated with the graph fusion algebra, and labelled by nodes $a,b\in G$ respectively. When combined with automorphisms, they lead to general integrable seams labelled by $x=(r,a,b,\kappa) \in (A_{g-2}, H, H, Z_2 )$ where H is the graph G itself for Type I theories and its parent for Type II theories. Identifying our construction labels with the conformal labels of Petkova and Zuber, we find that the integrable seams are in one-to-one correspondence with the conformal seams. The distinct seams are thus associated with the nodes of the Ocneanu quantum graph. The quantum symmetries and twisted partition functions are checked numerically for $|G| \leq 6$. We also show, in the case of $D_{2l}$, that the non-commutativity of the Ocneanu algebra of seams arises because the automorphisms do not commute with the fusions.

Lattice approach to excited TBA boundary flows: tricritical Ising model

We show how a lattice approach can be used to derive Thermodynamic Bethe Ansatz (TBA) equations describing all excitations for boundary flows. The method is illustrated for a prototypical flow of the tricritical Ising model by considering the continuum scaling limit of the A4 lattice model with integrable boundaries. Fixing the bulk weights to their critical values, the integrable boundary weights admit two boundary fields ξ and η which play the role of the perturbing boundary fields ϕ1,3 and ϕ1,2 inducing the renormalization group flow between boundary fixed points. The excitations are completely classified in terms of (m,n) systems and quantum numbers but the string content changes by certain mechanisms along the flow. For our prototypical example, we identify these mechanisms and the induced map between the relevant finitized Virasoro characters. We also solve the boundary TBA equations numerically to determine the flows for the leading excitations.