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Register a new userMore on the O(n) model on random maps via nested loops: loops with bending energy
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Added by
J\\'er\\'emie Bouttier
Publication date
2012
fields
Physics
and research's language is
English
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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.
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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 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.
We compute the generating functions of a O(n) model (loop gas model) on a random lattice of any topology. On the disc and the cylinder, they were already known, and here we compute all the other topologies. We find that the generating functions (and the correlation functions of the lattice) obey the topological recursion, as usual in matrix models, i.e they are given by the symplectic invariants of their spectral curve.
We pursue the analysis of nesting statistics in the $O(n)$ loop model on random maps, initiated for maps with the topology of disks and cylinders in math-ph/1605.02239, here for arbitrary topologies. For this purpose we rely on the topological recursion results of math-ph/0910.5896 and math-ph/1303.5808 for the enumeration of maps in the $O(n)$ model. We characterize the generating series of maps of genus $g$ with $k$ marked points and $k$ boundaries and realizing a fixed nesting graph. These generating series are amenable to explicit computations in the loop model with bending energy on triangulations, and we characterize their behavior at criticality in the dense and in the dilute phase.
We consider planar quadrangulations with three marked vertices and discuss the geometry of triangles made of three geodesic paths joining them. We also study the geometry of minimal separating loops, i.e. paths of minimal length among all closed paths passing by one of the three vertices and separating the two others in the quadrangulation. We concentrate on the universal scaling limit of large quadrangulations, also known as the Brownian map, where pairs of geodesic paths or minimal separating loops have common parts of non-zero macroscopic length. This is the phenomenon of confluence, which distinguishes the geometry of random quadrangulations from that of smooth surfaces. We characterize the universal probability distribution for the lengths of these common parts.
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