Dense Loop Model Research Articles

Critical site percolation on the triangular lattice: from integrability to conformal partition functions

Critical site percolation on the triangular lattice is described by the Yang–Baxter solvable dilute loop model with crossing parameter specialized to , corresponding to the contractible loop fugacity . We study the functional relations satisfied by the commuting transfer matrices of this model and the associated Bethe ansatz equations. The single and double row transfer matrices are respectively endowed with strip and periodic boundary conditions, and are elements of the ordinary and periodic dilute Temperley–Lieb algebras. The standard modules for these algebras are labelled by the number of defects d and, in the latter case, also by the twist . Nonlinear integral equation techniques are used to analytically solve the Bethe ansatz functional equations in the scaling limit for the central charge c = 0 and conformal weights . For the ground states, we find for strip boundary conditions and for periodic boundary conditions, where . We give explicit conjectures for the scaling limit of the trace of the transfer matrix in each standard module. For , these conjectures are supported by numerical solutions of the logarithmic form of the Bethe ansatz equations for the leading 20 or more conformal eigenenergies. With these conjectures, we apply the Markov traces to obtain the conformal partition functions on the cylinder and torus. These precisely coincide with our previous results for critical bond percolation on the square lattice, described by the dense loop model with . The concurrence of all this conformal data provides compelling evidence supporting a strong form of universality between these two stochastic models as logarithmic conformal field theories.

Exact densities of loops in O(1) dense loop model and of clusters in critical percolation on a cylinder: II. Rotated lattice

This work continues the study started in Povolotsky (2021 J. Phys. A: Math. Theor. 54 22LT01), where the exact densities of loops in the O(1) dense loop model on an infinite strip of the square lattice with periodic boundary conditions were obtained. These densities are also equal to the densities of critical percolation clusters on the 45∘ rotated square lattice rolled into a cylinder. Here, we extend those results to the square lattice with a tilt. This in particular allows us to obtain the densities of critical percolation clusters on the cylinder of the square lattice of standard orientation extensively studied before. We obtain exact densities of contractible and non-contractible loops or equivalently the densities of critical percolation clusters, which do not and do wrap around the cylinder, respectively. The solution uses the mapping of O(1) dense loop model to the six-vertex model in the Razumov–Stroganov point, while the effective tilt is introduced via the inhomogeneous transfer matrix proposed by Fujimoto. The further solution is based on the Bethe ansatz and Fridkin–Stroganov–Zagier’s solution of Baxter’s T–Q equation. The results are represented in terms of the solution of two explicit systems of linear algebraic equations, which can be performed either analytically for small circumferences of the cylinder or numerically for larger ones. We present exact rational values of the densities on the cylinders of small circumferences and several lattice orientations and use the results of high precision numerical calculations to study the finite-size corrections to the densities, in particular their dependence on the tilt of the lattice.

Critical behaviour of loop models on causal triangulations

We introduce a dense and a dilute loop model on causal dynamical triangulations. Both models are characterised by a geometric coupling constant g and a loop parameter α in such a way that the purely geometric causal triangulation model is recovered for α = 1. We show that the dense loop model can be mapped to a solvable planar tree model, whose partition function we compute explicitly and use to determine the critical behaviour of the loop model. The dilute loop model can likewise be mapped to a planar tree model; however, a closed-form expression for the corresponding partition function is not obtainable using the standard methods employed in the dense case. Instead, we derive bounds on the critical coupling g c and apply transfer matrix techniques to examine the critical behaviour for α small.

Exact densities of loops in O(1) dense loop model and of clusters in critical percolation on a cylinder

We obtain exact densities of contractible and non-contractible loops in the O(1) model on a strip of the square lattice rolled into an infinite cylinder of finite even circumference L. They are also equal to the densities of critical percolation clusters on 45 degree rotated square lattice rolled into a cylinder, which do not or do wrap around the cylinder respectively. The results are presented as explicit rational functions of L taking rational values for any even L. Their asymptotic expansions in the large L limit have irrational coefficients reproducing the earlier results in the leading orders. The solution is based on a mapping to the six-vertex model and the use of technique of Baxter’s T–Q equation.

The action of the Virasoro algebra in the two-dimensional Potts and loop models at generic Q

The spectrum of conformal weights for the CFT describing the two-dimensional critical Q-state Potts model (or its close cousin, the dense loop model) has been known for more than 30 years [1]. However, the exact nature of the corresponding Vir ⊗ overline{mathrm{Vir}} representations has remained unknown up to now. Here, we solve the problem for generic values of Q. This is achieved by a mixture of different techniques: a careful study of “Koo-Saleur generators” [2], combined with measurements of four-point amplitudes, on the numerical side, and OPEs and the four-point amplitudes recently determined using the “interchiral conformal bootstrap” in [3] on the analytical side. We find that null-descendants of diagonal fields having weights (hr,1, hr,1) (with r ∈ ℕ*) are truly zero, so these fields come with simple Vir ⊗ overline{mathrm{Vir}} (“Kac”) modules. Meanwhile, fields with weights (hr,s, hr,−s) and (hr,−s, hr,s) (with r, s ∈ ℕ*) come in indecomposable but not fully reducible representations mixing four simple Vir ⊗ overline{mathrm{Vir}} modules with a familiar “diamond” shape. The “top” and “bottom” fields in these diamonds have weights (hr,−s, hr,−s), and form a two-dimensional Jordan cell for L0 and {overline{L}}_0 . This establishes, among other things, that the Potts-model CFT is logarithmic for Q generic. Unlike the case of non-generic (root of unity) values of Q, these indecomposable structures are not present in finite size, but we can nevertheless show from the numerical study of the lattice model how the rank-two Jordan cells build up in the infinite-size limit.

Towers of Solutions of qKZ Equations and Their Applications to Loop Models

Cherednik’s type A quantum affine Knizhnik–Zamolodchikov (qKZ) equations form a consistent system of linear q-difference equations for V_n-valued meromorphic functions on a complex n-torus, with V_n a module over the mathrm{GL}_n-type extended affine Hecke algebra {mathcal {H}}_n. The family ({mathcal {H}}_n)_{nge 0} of extended affine Hecke algebras forms a tower of algebras, with the associated algebra morphisms {mathcal {H}}_nrightarrow {mathcal {H}}_{n+1}, in the Hecke algebra descending of arc insertion at the affine braid group level. In this paper, we consider qKZ towers (f^{(n)})_{nge 0} of solutions, which consist of twisted-symmetric polynomial solutions f^{(n)} (nge 0) of the qKZ equations that are compatible with the tower structure on ({mathcal {H}}_n)_{nge 0}. The compatibility is encoded by the so-called braid recursion relations: f^{(n+1)}(z_1,ldots ,z_{n},0) is required to coincide up to a quasi-constant factor with the push-forward of f^{(n)}(z_1,ldots ,z_{n}) by an intertwiner mu _{n}{:},V_{n}rightarrow V_{n+1} of {mathcal {H}}_{n}-modules, where V_{n+1} is considered as an {mathcal {H}}_{n}-module through the tower structure on ({mathcal {H}}_n)_{nge 0}. We associate with the dense loop model on the half-infinite cylinder with nonzero loop weights, a qKZ tower (f^{(n)})_{nge 0} of solutions. The solutions f^{(n)} are constructed from specialized dual non-symmetric Macdonald polynomials with specialized parameters using the Cherednik–Matsuo correspondence. In the special case that the extended affine Hecke algebra parameter is a third root of unity, f^{(n)} coincides with the (suitably normalized) ground state of the inhomogeneous dense O(1) loop model on the half-infinite cylinder with circumference n.

Extended T-systems, Q matrices and T-Q relations for models at roots of unity

The mutually commuting fused single and double-row transfer matrices of the critical six-vertex model are considered at roots of unity with crossing parameter a rational fraction of . The transfer matrices of the dense loop model analogs, namely the logarithmic minimal models , are similarly considered. For these models, we find explicit closure relations for the T-system functional equations and obtain extended sets of bilinear T-system identities. We also define extended Q matrices as linear combinations of the fused transfer matrices and obtain extended matrix T-Q relations. These results hold for diagonal twisted boundary conditions on the cylinder as well as invariant/Kac vacuum and off-diagonal/Robin vacuum boundary conditions on the strip. Using our extended T-system and extended T-Q relations for eigenvalues, we deduce the usual scalar Baxter T-Q relation and the Bazhanov–Lukyanov–Zamolodchikov decomposition of the fused transfer matrices and , at fusion level , in terms of the product or Q(u)2. It follows that the zeros of and are comprised of the Bethe roots and complete strings. We also clarify the formal observations of Pronko and Yang–Nepomechie–Zhang and establish, under favourable conditions, the existence of an infinite fusion limit in the auxiliary space of the fused transfer matrices. Despite this connection, the infinite-dimensional oscillator representations are not needed at roots of unity due to finite closure of the functional equations.

Two-point boundary correlation functions of dense loop models

We investigate six types of two-point boundary correlation functions in the dense loop model. These are defined as ratios Z/Z^0Z/Z0 of partition functions on the m\times nm×n square lattice, with the boundary condition for ZZ depending on two points xx and yy. We consider: the insertion of an isolated defect (a) and a pair of defects (b) in a Dirichlet boundary condition, the transition (c) between Dirichlet and Neumann boundary conditions, and the connectivity of clusters (d), loops (e) and boundary segments (f) in a Neumann boundary condition.For the model of critical dense polymers, corresponding to a vanishing loop weight (\beta = 0β=0), we find determinant and pfaffian expressions for these correlators. We extract the conformal weights of the underlying conformal fields and find \Delta = -\frac18Δ=−18, 00, -\frac3{32}−332, \frac3838, 11, \tfrac \theta \pi (1+ \tfrac{2\theta}\pi)θπ(1+2θπ), where \thetaθ encodes the weight of one class of loops for the correlator of type f. These results are obtained by analysing the asymptotics of the exact expressions, and by using the Cardy-Peschel formula in the case where xx and yy are set to the corners. For type b, we find a \ln|x-y|ln|x−y| dependence from the asymptotics, and a \ln (\ln n)ln(lnn) term in the corner free energy. This is consistent with the interpretation of the boundary condition of type b as the insertion of a logarithmic field belonging to a rank two Jordan cell.For the other values of \beta = 2 \cos \lambdaβ=2cosλ, we use the hypothesis of conformal invariance to predict the conformal weights and find \Delta = \Delta_{1,2}Δ=Δ1,2, \Delta_{1,3}Δ1,3, \Delta_{0,\frac12}Δ0,12, \Delta_{1,0}Δ1,0, \Delta_{1,-1}Δ1,−1 and \Delta_{\frac{2\theta}\lambda+1,\frac{2\theta}\lambda+1}Δ2θλ+1,2θλ+1, extending the results of critical dense polymers. With the results for type f, we reproduce a Coulomb gas prediction for the valence bond entanglement entropy of Jacobsen and Saleur.

Asymptotics of symmetric polynomials with applications to statistical mechanics and representation theory

We develop a new method for studying the asymptotics of symmetric polynomials of representation-theoretic origin as the number of variables tends to infinity. Several applications of our method are presented: We prove a number of theorems concerning characters of infinite-dimensional unitary group and their $q$-deformations. We study the behavior of uniformly random lozenge tilings of large polygonal domains and find the GUE-eigenvalues distribution in the limit. We also investigate similar behavior for alternating sign matrices (equivalently, six-vertex model with domain wall boundary conditions). Finally, we compute the asymptotic expansion of certain observables in $O(n=1)$ dense loop model.

The Toggle Group, Homomesy, and the Razumov-Stroganov Correspondence

The Razumov-Stroganov correspondence, an important link between statistical physics and combinatorics proved in 2011 by L. Cantini and A. Sportiello, relates the ground state eigenvector of the $O(1)$ dense loop model on a semi-infinite cylinder to a refined enumeration of fully-packed loops, which are in bijection with alternating sign matrices. This paper reformulates a key component of this proof in terms of posets, the toggle group, and homomesy, and proves two new homomesy results on general posets which we hope will have broader implications.

A one-parameter refinement of the Razumov–Stroganov correspondence

We introduce and prove a one-parameter refinement of the Razumov–Stroganov correspondence. This is achieved for fully-packed loop configurations (FPL) on domains which generalise the square domain, and which are endowed with the gyration operation. We consider one given side of the domain, and FPLs such that the only straight-line tile on this side is black. We show that the enumeration vector associated with such FPLs, weighted according to the position of the straight line and refined according to the link pattern for the black boundary points, is the ground state of the scattering matrix, an integrable one-parameter deformation of the O(1) Dense Loop Model Hamiltonian. We show how the original Razumov–Stroganov correspondence, and a conjecture formulated by Di Francesco in 2004, follows from our results.

Integrability as a consequence of discrete holomorphicity: loop models

In this paper, we explore the relationship between integrability and the discrete holomorphicity of a class of complex lattice observables in the context of the Potts dense loop model and the O(n) dilute loop model. It is shown that the conditions for integrability, namely, the inversion and Yang–Baxter relations, can be derived from the condition of holomorphicity of the observables. Furthermore, the Z-invariance of the models is shown to result in the invariance of the observables on the boundary of a sublattice under reshuffling of the rhombuses of its planar rhombic embedding.

Inhomogeneous loop models with open boundaries

We consider the crossing and non-crossing O(1) dense loop models on a semi-infinite strip, with inhomogeneities (spectral parameters) that preserve the integrability. We compute the components of the ground state vector and obtain a closed expression for their sum, in the form of Pfaffian and determinantal formulas.

Folding the square-diagonal lattice

We study the problem of “phantom” folding of the two-dimensional square lattice, in which the edges and diagonals of each face can be folded. The nonvanishing thermodynamic folding entropy per face s ≅ .2299(1) is estimated both analytically and numerically, by successively mapping the model onto a dense loop model, a spin model and a new 28 Vertex, 4-color model. Higher-dimensional generalizations are investigated, as well as other foldable lattices. © 1998 Elsevier Science B.V

Multi-colour braid-monoid algebras

We define multi-colour generalizations of braid-monoid algebras and present explicit matrix representations which are related to two-dimensional exactly solvable lattice models of statistical mechanics. In particular, we show that the two-colour braid-monoid algebra describes the Yang-Baxter algebra of the critical dilute A-D-E models which were recently introduced by Warnaar, Nienhuis, and Seaton as well as by Roche. These and other solvable models related to dense and dilute loop models are discussed in detail and it is shown that the solvability is a direct consequence of the algebraic structure. It is conjectured that the Yang-Baxterization of general multi-colour braid-monoid algebras will lead to the construction of further solvable lattice models.