fields also have some combinatorial interpretations, and have been studied for various applications. **The** most famous is **the** Kontsevich integral, which is **the** generating function for intersection numbers [33, 24, 19].
Here, we solve this more general model.
Multimatrix model also play an important role in quantum gravity and string theory, where they play **the** role **of** a regularized discrete space-time. **The** 1-matrix model, counts discrete surfaces without color, and is a model for quantum gravity without matter, whereas **the** **chain** **of** **matrices** counts discrete surfaces with n colors, and is interpreted as a model **of** quantum gravity with some matter field [27, 10, 1, 29, 30, 13], namely a matter which can have n possible states. More recently, matrix models have played a role in **topological** string theory [12].

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1 Introduction
Asymptotic expansions in Random Matrix Theory created bridges between different worlds, including topology, statistical mechanics, and quantum field theory. In mathematics, a breakthrough was made in 1986 by Harer and Zagier who used **the** large dimension **expansion** **of** **the** moments **of** Gaussian **matrices** to compute **the** Euler characteristic **of** **the** moduli space **of** curves. A good introduction to this topic is given in **the** survey [32] by Zvonkin. In physics, **the** seminal works **of** t’Hooft [28] and Brézin, Parisi, Itzykson and Zuber [20] related matrix models with **the** enumeration **of** maps **of** any genus, hence providing a purely analytical tool to solve these hard combinatorial problems. Considering **matrices** in interaction via a potential, **the** so-called matrix models, indeed allows to consider **the** enumeration **of** maps with several vertices, including a possible coloring **of** **the** edges when **the** matrix model contains several **matrices**. This relation allowed to associate matrix models to statistical models on random graphs [38, 11, 23, 24, 26], as well as in [19] and [25] for **the** unitary case. This was also extended to **the** so-called β-ensembles in [27, 17, 12, 13, 15, 16]. Among other objects, these works study correlation functions and **the** so-called free energy and show that they expand as power series in **the** inverse **of** **the** dimension, and **the** coefficients **of** these expansions enumerate maps sorted by their genus. To compute asymptotic expansions, often referred to in **the** literature as **topological** expansions, one **of** **the** most successful methods is **the** loop equations method, see [35] and [36]. Depending on **the** model **of** random matrix, those are Tutte’s equations, Schwinger-Dyson equations, Ward identities, Virasoro constraints, W-algebra or simply integration by part. This method was refined and used repeatedly in physics, see for example **the** work **of** Eynard and his collaborators, [21, 22, 18, 14]. At first those equations were only solved for **the** first few orders, however in 2004, in [22] and later [33] and [34], this method was refined to push **the** **expansion** to any orders recursively [37].

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CNRS, URA 2306, F-91191 Gif-sur-Yvette, France.
Abstract
We solve **the** loop equations **of** **the** β-ensemble model analogously to **the** solution found for **the** Hermitian **matrices** β = 1. For β = 1, **the** solution was expressed using **the** algebraic spectral curve **of** equation y 2 = U (x). For arbitrary β, **the** spectral curve converts into a Schr¨ odinger equation ((~∂) 2 −U(x))ψ(x) = 0 with ~ ∝ ( √ β −1/ √ β)/N . This paper is similar to **the** sister paper I, in particular, all **the** main ingredients specific for **the** algebraic solution **of** **the** problem remain **the** same, but here we present **the** second approach to finding a solution **of** loop equations using sectorwise definition **of** resolvents. Being technically more involved, it allows to define consistently **the** B-cycle structure **of** **the** obtained quantum algebraic curve (a D-module **of** **the** form y 2 −U(x), where [y, x] = ~) and to construct explicitly **the** correlation functions and **the** corresponding symplectic invariants F h , or **the** terms **of** **the** free energy, in 1/N 2 -**expansion** at arbitrary ~.

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rank (or equivalently is invertible), then there exists an open set U ⊂ R n+1
containing a such that there exists a unique continuously differentiable function g : U → R d−n such that g(a) = b and F (x, g(x)) = 0 for all x ∈ U .
To prove **the** existence **of** **the** isotopy from f −1 (0) to f P L −1 (0), we will apply **the** implicit function theorem (Theorem 6) to several functions g that are close to f and we will therefore need to prove that their Jacobians are **of** maximal rank. A matrix has maximal rank if and only if **the** Gram matrix **of** its columns has a non-zero determinant or, equivalently, non-zero eigenvalues. In our context, we will need lower bounds on **the** absolute values **of** **the** eigenvalues **of** **the** Gram **matrices** Gram(∇g), given **the** lower bound λ min on **the** absolute

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This natural generalization **of** diagrammatic techniques from **the** 1MM to **the** 2MM also points out that **the** study **of** **the** **chain** **of** **matrices** [21] would give a more general knowledge **of** **the** link between algebraic geometry and matrix models.
Let us also mention that we have considered here only polynomial potentials, but it is clear that **the** whole method should extend easily to **the** more general class **of** semi-classical potentials [6, 7], whose loop equations are very similar [23]. **The** 1-MM with hard edges was already treated in [12].

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For **the** converse, we use **the** proof **of** Theorem 3.1: we have ξ = M · ξ, where M is a product **of** **matrices** L and R; this product has L’s and R’s in it. We take **the** corresponding product f **of** morphisms (a 7→ a, b 7→ ab) and (a 7→ ab, b 7→ b), whose abelianizations are R and L. Then f sends a onto au, u 6= 1, and therefore f has a fixpoint s, which is **the** limit **of** **the** finite words f n (a) (each **of** which is a proper prefix **of** **the** next one). Then by (5.2) **the** slope **of** s is ξ. **The** infinite word s is Sturmian, since **the** two previous morphisms are Sturmian, so that f n (a) is balanced and since **the** slope **of** s is irrational, see [22, Prop. 2.1.11 and Thm. 2.1.5].

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This quantity does not depend on j, and f is clearly a meromorphic 1-form, whose poles can be easily seen on this expression using **the** recursion hypothesis.
**The** fact that **the** A and B cycle integrals vanish comes from **the** symmetry x ↔ y. Indeed under **the** symmetry x ↔ y, f is changed to ˜ f and ˜ f is changed to f . At **the** same time **the** A-cycles are changed to −A because 2iπǫ = H A ydx = − H A xdy, and **the** B-cycles are changed to −B in order to form a canonical basis. Therefore, **the** A and B cycle integrals **of** f + ˜ f vanish.

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We also show that Z N (E) satisfies bilinear Hirota equations, and thus Z N (E) is a
formal τ -function, and we construct **the** associated formal Baker-Hakiezer function [9]. We thus have a notion **of** a τ function associated to an algebraic curve. Such notion has already been encountered in **the** litterature [9], and it is not clear whether our definition coincides with other existing definitions. What can be understood so far, is that we are defining a sort **of** quantum deformation **of** a classical τ -function whose spectral curve is E. **The** classical τ function being only **the** dispersionless limit ln Z ∞ (E) = −F (0) (E), while our Z N (E) concerns **the** full system.

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Math´ ematiques, Montr´ eal. 2
? Institut de physique th´ eorique, CEA Saclay, France. 3
Abstract: **The** goal **of** this article is to rederive **the** connection between **the** Painlev´ e 5 integrable system and **the** universal eigenvalues correlation functions **of** double-scaled hermi- tian matrix models, through **the** **topological** recursion method. More specifically we prove, to all orders, that **the** WKB asymptotic expansions **of** **the** τ -function as well as **of** determinan- tal formulas arising from **the** Painlev´ e 5 Lax pair are identical to **the** large N double scaling asymptotic expansions **of** **the** partition function and correlation functions **of** any hermitian matrix model around a regular point in **the** bulk. In other words, we rederive **the** “sine-law” universal bulk asymptotic **of** large random **matrices** and provide an alternative perturbative proof **of** universality in **the** bulk with only algebraic methods. Eventually we exhibit **the** first orders **of** **the** series **expansion** up to O(N −5 ).

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Most often **the** spectral curve happens to be a very simple and natural geometric object from **the** A-model point **of** view. It is typically **the** ”most probable shape” **of** **the** objects counted in **the** A-model, in some ”large size limit”. This is **the** case for random **matrices**, **the** spectral curve is **the** large size limit **of** **the** eigenvalue density. Another example occurs in counting plane partitions, where **the** spectral curve is **the** shape **of** **the** limiting plane partition (often called arctic circle). But there is unfortunately no general recipe **of** how to ﬁnd **the** spectral curve mirror **of** a given A-model.

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Interestingly, crossover between **the** symplectic and unitary universality classes upon breaking time-reversal symmetry has been observed in ultrathin samples **of** Bi 2 Te 3 [ 16 ].
Theoretically, describing **the** transport properties **of** TI SSs amounts to considering **the** diffusion properties **of** two-dimensional (2D) Dirac fermions. Indeed, at **the** lowest order in k .p theory **the** SS Fermi surface is circular with a spin winding in **the** plane. At low energy, this 2D conductor shares a number **of** similarities with graphene, but with **the** following important differences: **the** momentum is locked to a real spin as opposed to an A–B sublattice pseudo-spin in graphene, and it has a single Dirac cone as opposed to **the** fourfold degeneracy **of** **the** Dirac cone in graphene. For this simpler Dirac metal, **the** conductivity and **the** induced in-plane spin polarization were calculated as functions **of** **the** 2D carrier concentration [ 17 ]. Far from **the** Dirac point, surface spin–orbit may generically produce a significant hexagonal warping (HW) **of** **the** spin texture. As a result, **the** Fermi surface exhibits a snowflake or nearly hexagonal Fermi surface depending on **the** carrier density, and **the** spin gets tilted out **of** **the** plane [ 18 ]. Those effects have been confirmed by ARPES and scanning tunneling microscopy experiments performed on Bi 2 Te 3 crystals, where HW is particularly strong [ 19 – 23 ], and also

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We prove Theorem 2 using Theorem 1 as an intermediate step. This strategy is inspired by [4] in their study **of** Schr ¨odinger operators. In contrast to **the** result in [4], **the** density **of** mode-locking for dynamically forced maps is true regardless **of** **the** range **of** **the** Schwartzman asymptotic cycle G ( g ) (for its definition, see [21] or [4, Section 1.1]), while a SL ( 2, R )− cocycle f over base map g can be mode- locked only if ρ ( f ) ∈ G ( g ) mod Z. This is due to **the** fact that for dynamically forced maps, one can perform perturbations that only act locally within **the** fibers, a convenient feature that is not shared by SL ( 2, R )− cocycles.

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tion **of** this bundle in terms **of** cocycles and tries to make **the** links with **the** strictly **topological** presentation (representation **of** **the** fundamental group) proposed by Arnol’d [3], originally in an appendix **of** **the** book **of** Maslov [12]. This link is established only for **the** lagrangian submanifolds
**of** T ∗ R n . I propose in this work a new construction (1.2) for **the** lagrangian submanifolds **of** T ∗ X,

imal ideal m = (7r) and finite residue field R/1rR. We also assume that A is a quaternion K-algebra. Let q be **the** number **of** elements in R/1rR.
For simplicity, **the** number **of** principal left ideals in A whose norm is equal
to (1rm) will be denoted by L(A, m) (later we return to **the** notation from Section 1: t(A, mm)). **The** following result is well-known (see e.g. [9], §2):

Our next results concern **the** indepence **of** different digital expansions.
For example, in [6] **the** following property is shown. Suppose that ql, q2 are
two coprime integers and fl, f2 qm resp. q2-additive functions satisfying **the**
assumptions **of** Theorem 1. Then we have, as N --> oo,