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Register a new userLoop models on random maps via nested loops: case of domain symmetry breaking and application to the Potts model
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We use the nested loop approach to investigate loop models on random planar maps where the domains delimited by the loops are given two alternating colors, which can be assigned different local weights, hence allowing for an explicit Z_2 domain symmetry breaking. Each loop receives a non local weight n, as well as a local bending energy which controls loop turns. By a standard cluster construction that we review, the Q = n^2 Potts model on general random maps is mapped to a particular instance of this problem with domain-non-symmetric weights. We derive in full generality a set of coupled functional relations for a pair of generating series which encode the enumeration of loop configurations on maps with a boundary of a given color, and solve it by extending well-known complex analytic techniques. In the case where loops are fully-packed, we analyze in details the phase diagram of the model and derive exact equations for the position of its non-generic critical points. In particular, we underline that the critical Potts model on general random maps is not self-dual whenever Q eq 1. In a model with domain-symmetric weights, we also show the possibility of a spontaneous domain symmetry breaking driven by the bending energy.
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We continue our investigation of the nested loop approach to the O(n) model on random maps, by extending it to the case where loops may visit faces of arbitrary degree. This allows to express the partition function of the O(n) loop model as a specialization of the multivariate generating function of maps with controlled face degrees, where the face weights are determined by a fixed point condition. We deduce a functional equation for the resolvent of the model, involving some ring generating function describing the immediate vicinity of the loops. When the ring generating function has a single pole, the model is amenable to a full solution. Physically, such situation is realized upon considering loops visiting triangles only and further weighting these loops by some local bending energy. Our model interpolates between the two previously solved cases of triangulations without bending energy and quadrangulations with rigid loops. We analyze the phase diagram of our model in details and derive in particular the location of its non-generic critical points, which are in the universality classes of the dense and dilute O(n) model coupled to 2D quantum gravity. Similar techniques are also used to solve a twisting loop model on quadrangulations where loops are forced to make turns within each visited square. Along the way, we revisit the problem of maps with controlled, possibly unbounded, face degrees and give combinatorial derivations of the one-cut lemma and of the functional equation for the resolvent.
We consider the O(n) loop model on tetravalent maps and show how to rephrase it into a model of bipartite maps without loops. This follows from a combinatorial decomposition that consists in cutting the O(n) model configurations along their loops so that each elementary piece is a map that may have arbitrary even face degrees. In the induced statistics, these maps are drawn according to a Boltzmann distribution whose parameters (the face weights) are determined by a fixed point condition. In particular, we show that the dense and dilute critical points of the O(n) model correspond to bipartite maps with large faces (i.e. whose degree distribution has a fat tail). The re-expression of the fixed point condition in terms of linear integral equations allows us to explore the phase diagram of the model. In particular, we determine this phase diagram exactly for the simplest version of the model where the loops are rigid. Several generalizations of the model are discussed.
We compute the partition function of the $q$-states Potts model on a random planar lattice with $pleq q$ allowed, equally weighted colours on a connected boundary. To this end, we employ its matrix model representation in the planar limit, generalising a result by Voiculescu for the addition of random matrices to a situation beyond free probability theory. We show that the partition functions with $p$ and $q-p$ colours on the boundary are related algebraically. Finally, we investigate the phase diagram of the model when $0leq qleq 4$ and comment on the conformal field theory description of the critical points.
We extend a recent analysis of the $q$-states Potts model on an ensemble of random planar graphs with $pleqslant q$ allowed, equally weighted, spins on a connected boundary. In this paper we explore the $(q<4,pleqslant q)$ parameter space of finite-sheeted resolvents and derive the associated critical exponents. By definition a value of $q$ is allowed if there is a $p=1$ solution, and we reproduce the long-known result that $q= 2(1+cos{frac{m}{n} pi})$ with $m,n$ coprime. In addition we find that there are two distinct sequences of solutions, one of which contains $p=2$ and $p=q/2$ while the other does not. The boundary condition $p=3$ appears only for $q=3$ which also has a $p=3/2$ boundary condition; we conjecture that this new solution corresponds in the scaling limit to the New boundary condition, discovered on the flat lattice by Affleck et al. We also explore Kramers-Wannier duality for $q=3$ in this context and explicitly construct the known boundary conditions; we show that the mixed boundary condition is dual to a boundary condition on dual graphs that corresponds to Affleck et als identification of the New boundary condition on fixed lattices. On the other hand we find that the mixed boundary condition of the dual, and the corresponding New boundary condition of the original theory are not described by conventional resolvents.
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