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This article is the last of four that completely and rigorously characterize a solution space $${{\mathcal{S}}_N}$$ for a homogeneous system of 2 N + 3 linear partial differential equations in 2 N variables that arises in conformal field theory (CFT) and multiple Schramm-Löwner evolution (SLE $${_\kappa}$$ ). The system comprises 2 N null-state equations and three conformal Ward identities that govern CFT correlation functions of 2 N one-leg boundary operators. In the first two articles (Flores and Kleban in Commun Math Phys, ; Flores and Kleban, in Commun Math Phys, ), we use methods of analysis and linear algebra to prove that dim $${{\mathcal{S}}_N \leq C_N}$$ , with C the Nth Catalan number. Using these results in the third article (Flores and Kleban, in Commun Math Phys, ), 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 $${8/\kappa}$$ is odd, in which case a logarithmic term may appear). This establishes an important element in the operator product expansion 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 $${_\kappa}$$ process join in a particular connectivity. This leads to new formulas for crossing probabilities of critical lattice models inside polygons with a free/fixed side-alternating boundary condition, which we derive in Flores et al. (Partition functions and crossing probabilities for critical systems inside polygons, in preparation). Finally, we propose a reason for why the exceptional speeds [certain $${\kappa}$$ values that appeared in the analysis of the Coulomb gas solutions in Flores and Kleban (Commun Math Phys, )] and the minimal models of CFT are connected. [ABSTRACT FROM AUTHOR]

This article is the third of four that completely and rigorously characterize a solution space $${\mathcal{S}_N}$$ for a homogeneous system of 2 N + 3 linear partial differential equations (PDEs) in 2 N variables that arises in conformal field theory (CFT) and multiple Schramm-Löwner evolution (SLE). The system comprises 2 N null-state equations and three conformal Ward identities that govern CFT correlation functions of 2 N one-leg boundary operators. In the first two articles (Flores and Kleban, in Commun Math Phys, , ; Commun Math Phys, , ), we use methods of analysis and linear algebra to prove that dim $${\mathcal{S}_N\leq C_N}$$ , with C the Nth 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 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 Flores and Kleban (Commun Math Phys, , ) 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 $${C_N\times C_N}$$ matrix, with the form of a meander matrix. If the determinant of this matrix does not vanish, then the set of C Coulomb gas solutions is linearly independent. And if this determinant does vanish, then we construct an alternative set of C 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. We emphasize that, although the system of PDEs arises in CFT in away that is typically non-rigorous, our treatment of this system here and in Flores and Kleban (Commun Math Phys, , ; Commun Math Phys, , ; Commun Math Phys, , ) is completely rigorous. [ABSTRACT FROM AUTHOR]

This article is the second of four that completely and rigorously characterize a solution space $${\mathcal{S}_N}$$ for a homogeneous system of 2 N + 3 linear partial differential equations in 2 N variables that arises in conformal field theory (CFT) and multiple Schramm-Löwner evolution (SLE $${_\kappa}$$ ). The system comprises 2 N null-state equations and three conformal Ward identities which govern CFT correlation functions of 2 N one-leg boundary operators. In the first article (Flores and Kleban, Commun Math Phys, , ), we use methods of analysis and linear algebra to prove that dim $${\mathcal{S}_N \leq C_N}$$ , with C the Nth Catalan number. The analysis of that article is complete except for the proof of a lemma that it invokes. The purpose of this article is to provide that proof. The lemma states that if every interval among ( x, x), ( x, x),...,( x, x) is a two-leg interval of $${F \in \mathcal{S}_N}$$ (defined in Flores and Kleban, Commun Math Phys, , ), 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. Estimates in which two interval lengths vanish simultaneously divide into two cases: two adjacent intervals and two non-adjacent intervals. The analysis of the latter case is similar to that for one vanishing interval length. In contrast, the analysis of the former case is more complicated, involving a Green function that contains the Jacobi heat kernel as its essential ingredient. [ABSTRACT FROM AUTHOR]

This article is the first of four that completely and rigorously characterize a solution space $${{\mathcal{S}}_N}$$ for a homogeneous system of 2 N + 3 linear partial differential equations (PDEs) in 2 N variables that arises in conformal field theory (CFT) and multiple Schramm-Löwner evolution (SLE $${_\kappa}$$ ). 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-gas model, such as percolation, or more generally the Potts models and O( n) models, at the statistical mechanical critical point. (SLE $${_\kappa}$$ partition functions also satisfy these equations.) For such a lattice model in a polygon $${{\mathcal{P}}}$$ with its 2 N sides exhibiting a free/fixed side-alternating boundary condition $${\vartheta}$$ , this partition function is proportional to the CFT correlation function where the w are the vertices of $${{\mathcal{P}}}$$ and where $${\psi_1^c}$$ is a one-leg corner operator. (Partition functions for 'crossing events' in which clusters join the fixed sides of $${{\mathcal{P}}}$$ in some specified connectivity are linear combinations of such correlation functions.) When conformally mapped onto the upper half-plane, methods of CFT show that this correlation function satisfies the system of PDEs that we consider. In this first article, we use methods of analysis to prove that the dimension of this solution space is no more than C, the Nth Catalan number. While our motivations are based in CFT, our proofs are completely rigorous. This proof is contained entirely within this article, except for the proof of Lemma 14, which constitutes the second article (Flores and Kleban, in Commun Math Phys, , ). In the third article (Flores and Kleban, in Commun Math Phys, , ), we use the results of this article to prove that the solution space of this system of PDEs has dimension C and is spanned by solutions constructed with the CFT Coulomb gas (contour integral) formalism. In the fourth article (Flores and Kleban, in Commun Math Phys, , ), we prove further CFT-related properties about these solutions, some useful for calculating cluster-crossing probabilities of critical lattice models in polygons. [ABSTRACT FROM AUTHOR]

We generalize the number theoretic spin chain, a one-dimensional statistical model based on the Farey fractions, by introducing a parameter $$x \geqslant 0$$ . This allows us to write recursion relations in the length of the chain. These relations are closely related to the Lewis three-term equation, which is useful in the study of the Selberg ζ-function. We then make use of these relations and spin orientation transformations. We find a simple connection with the transfer operator of a model of intermittency in dynamical systems. In addition, we are able to calculate certain spin expectation values explicitly in terms of the free energy or correlation length. Some of these expectation values appear to be directly connected with the mechanism of the phase transition [ABSTRACT FROM AUTHOR]

We consider the Farey fraction spin chain, a one-dimensional model defined on (the matrices generating) the Farey fractions. We extend previous work on the thermodynamics of this model by introducing an external field h. From rigorous and more heuristic arguments, we determine the phase diagram and phase transition behavior of the extended model. Our results are fully consistent with scaling theory (for the case when a “marginal” field is present) despite the unusual nature of the transition for h = 0, and the presence of long-range forces. [ABSTRACT FROM AUTHOR]

We examine crossing probabilities and free energies for conformally invariant critical 2-D systems in rectangular geometries, derived via conformal field theory and Stochastic Löwner Evolution methods. These quantities are shown to exhibit interesting modular behavior, although the physical meaning of modular transformations in this context is not clear. We show that in many cases these functions are completely characterized by very simple transformation properties. In particular, Cardy's function for the percolation crossing probability (including the conformal dimension 1/3), follows from a simple modular argument. A new type of “higher-order modular form” arises and its properties are discussed briefly. [ABSTRACT FROM AUTHOR]

In previous work, two of the authors determined, completely and rigorously, a solution space S N for a homogeneous system of 2 N + 3 linear partial differential equations (PDEs) in 2 N variables that arises in conformal field theory (CFT) and multiple Schrammâ€"Löwner evolution (SLE Îş ). The system comprises 2 N null-state equations and three conformal ward identities that govern CFT correlation functions of 2 N one-leg boundary operators or SLE Îş partition functions. M Bauer et al conjectured a formula, expressed in terms of ‘pure SLE Îş partition functions,’ for the probability that the growing curves of a multiple-SLE Îş process join in a particular connectivity. In a previous article, we rigorously define certain elements of S N , which we call ‘connectivity weights,’ argue that they are in fact pure SLE Îş partition functions, and show how to find explicit formulas for them in terms of Coulomb gas contour integrals. Our formal definition of the connectivity weights immediately leads to a method for finding explicit expressions for them. However, this method gives very complicated formulas where simpler versions may be available, and it is not applicable for certain values of Îş ∈ (0, 8) corresponding to well-known critical lattice models in statistical mechanics. In this article, we determine expressions for all connectivity weights in S N for N ∈ {1, 2, 3, 4} (those with N ∈ {3, 4} are new) and for so-called ‘rainbow connectivity weights’ in S N for all N ∈ Z + + 1. We verify these formulas by explicitly showing that they satisfy the formal definition of a connectivity weight. In appendix, we investigate logarithmic singularities of some of these expressions, appearing for certain values of Îş predicted by logarithmic CFT. [ABSTRACT FROM AUTHOR]

We consider the Farey fraction spin chain in an external field h. Using ideas from dynamical systems and functional analysis, we show that the free energy f in the vicinity of the second-order phase transition is given, exactly, by Here $t=\lambda_{G}\log(2)(1-\frac{\beta}{\beta_{c}})$ is a reduced temperature, so that the deviation from the critical point is scaled by the Lyapunov exponent of the Gauss map, λ G. It follows that λ G determines the amplitude of both the specific heat and susceptibility singularities. To our knowledge, there is only one other microscopically defined interacting model for which the free energy near a phase transition is known as a function of two variables. Our results confirm what was found previously with a cluster approximation, and show that a clustering mechanism is in fact responsible for the transition. However, the results disagree in part with a renormalisation group treatment. [ABSTRACT FROM AUTHOR]

We consider the Farey fraction spin chain in an external field h. Utilising ideas from dynamical systems, the free energy of the model is derived by means of an effective cluster energy approximation. This approximation is valid for divergent cluster sizes, and hence appropriate for the discussion of the magnetizing transition. We calculate the phase boundaries and the scaling of the free energy. At h = 0 we reproduce the rigorously known asymptotic temperature dependence of the free energy. For h ≠ 0, our results are largely consistent with those found previously using mean field theory and renormalization group arguments. [ABSTRACT FROM AUTHOR]