Gaëtan Borot

A matrix model for simple Hurwitz numbers, and topological recursion

Abstract We introduce a new matrix model representation for the generating function of simple Hurwitz numbers. We calculate the spectral curve of the model and the associated symplectic invariants developed in Eynard and Orantin (2007) [3] . As an application, we prove the conjecture proposed by Bouchard and Marino (2008) [1] , relating Hurwitz numbers to the spectral invariants of the Lambert curve e x = y e − y .

All order asymptotics of hyperbolic knot invariants from non-perturbative topological recursion of A-polynomials

We propose a conjecture to compute the all-order asymptotic expansion of the colored Jones polynomial of the complement of a hyperbolic knot, J_N(q = exp(2u/N)) when N goes to infinity. Our conjecture claims that the asymptotic expansion of the colored Jones polynomial is a the formal wave function of an integrable system whose semiclassical spectral curve S would be the SL_2(C) character variety of the knot (the A-polynomial), and is formulated in the framework of the topological recursion. It takes as starting point the proposal made recently by Dijkgraaf, Fuji and Manabe (who kept only the perturbative part of the wave function, and found some discrepancies), but it also contains the non-perturbative parts, and solves the discrepancy problem. These non-perturbative corrections are derivatives of Theta functions associated to S, but the expansion is still in powers of 1/N due to the special properties of A-polynomials. We provide a detailed check for the figure-eight knot and the once-punctured torus bundle L^2R. We also present a heuristic argument inspired from the case of torus knots, for which knot invariants can be computed from a matrix model.

A recursive approach to the O(n) model on random maps via nested loops

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.

Enumeration of maps with self avoiding loops and the O(n) model on random lattices of all topologies

We compute the generating functions of an model (loop gas model) on a random lattice of any topology. On the disc and the cylinder, the topologies were already known, and here we compute all the other topologies. We find that they obey a slightly deformed version of the topological recursion valid for the 1-Hermitian matrix models. The generating functions of genus g maps without boundaries are given by the symplectic invariants Fg of a spectral curve . This spectral curve was known before, and it is in general not algebraic.

More on the O(n) model on random maps via nested loops: loops with bending energy

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 us 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 a 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.

RIGHT TAIL ASYMPTOTIC EXPANSION OF TRACY–WIDOM BETA LAWS

Using loop equations, we compute the large deviation function of the maximum eigenvalue to the right of the spectrum in the Gaussian β matrix ensembles, to all orders in 1/N. We then give a physical derivation of the all order asymptotic expansion of the right tail of Tracy–Widom β laws, for all β > 0, by studying the double scaling limit.

Rational Differential Systems, Loop Equations, and Application to the

To any solution of a linear system of differential equations, we associate a matrix kernel, correlators satisfying a set of loop equations, and in the presence of isomonodromic parameters, a Tau function. We then study their semiclassical expansion (WKB type expansion in powers of the weight

th Reductions of KP

{\hbar}