Haut PDF Topological expansion of the chain of matrices

Topological expansion of the chain of matrices

Topological expansion of the chain of matrices

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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Topological expansion of the Bethe ansatz, and non-commutative algebraic geometry

Topological expansion of the Bethe ansatz, and non-commutative algebraic geometry

topological strings [3]. In particular they are related to the Kodaira-Spencer field theory [8]. There were many attempts to compute also non-hermitian matrix integrals, and an attempt to extend the method of [13] was first made in [7], and here in this paper we deeply improve the result of [7]. The aim of the construction we present here, is to define F (g) ’s for a ”non-commutative spectral curve”, i.e. a non commutative

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Asymptotic expansion of smooth functions in polynomials in deterministic matrices and iid GUE matrices

Asymptotic expansion of smooth functions in polynomials in deterministic matrices and iid GUE matrices

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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Topological expansion of the 2-matrix model correlation functions: diagrammatic rules for a residue formula

Topological expansion of the 2-matrix model correlation functions: diagrammatic rules for a residue formula

The future prospects of that work are to find the diagrammatic rules for com- puting the free energy to all order in the topological expansion, and also all mixed correlation functions (using the result of [14]). Another possible extension is to work out the multimatrix model, i.e. the chain of matrices as in [20], and in particular the limit of matrix quantum mechanics. We believe that this technique could apply to many other integrable models.

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Topological expansion of the Bethe ansatz, and non-commutative algebraic geometry

Topological expansion of the Bethe ansatz, and non-commutative algebraic geometry

topological strings [3]. In particular they are related to the Kodaira-Spencer field theory [8]. There were many attempts to compute also non-hermitian matrix integrals, and an attempt to extend the method of [13] was first made in [7], and here in this paper we deeply improve the result of [7]. The aim of the construction we present here, is to define F (g) ’s for a ”non-commutative spectral curve”, i.e. a non commutative

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Topological expansion of the β-ensemble model and quantum algebraic geometry in the sectorwise approach

Topological expansion of the β-ensemble model and quantum algebraic geometry in the sectorwise approach

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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The topological correctness of PL-approximations of isomanifolds

The topological correctness of PL-approximations of isomanifolds

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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Free energy topological expansion for the 2-matrix model.

Free energy topological expansion for the 2-matrix model.

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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On the Stern–Brocot expansion of real numbers

On the Stern–Brocot expansion of real numbers

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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Topological expansion of mixed correlations in the hermitian 2 Matrix Model and x-y symmetry of the F_g invariants.

Topological expansion of mixed correlations in the hermitian 2 Matrix Model and x-y symmetry of the F_g invariants.

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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Invariants of algebraic curves and topological expansion

Invariants of algebraic curves and topological expansion

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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The sinelaw gap probability, Painlevé 5, and asymptotic expansion by the topological recursion

The sinelaw gap probability, Painlevé 5, and asymptotic expansion by the topological recursion

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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Topological expansion in isomorphisms with random walks for matrix valued fields

Topological expansion in isomorphisms with random walks for matrix valued fields

TOPOLOGICAL EXPANSION IN ISOMORPHISMS WITH RANDOM WALKS FOR MATRIX VALUED FIELDS TITUS LUPU Abstract. We consider Gaussian fields of real symmetric, complex Hermitian or quaternionic Hermitian matrices over an electrical network, and describe how the isomorphisms with random walks for these fields make appear topological expansions encoded by ribbon graphs. We further consider matrix valued Gaussian fields twisted by an orthogonal, unitary or symplectic connection. In this case the isomorphisms make appear traces of holonomies of the connection along random walk loops parametrized by border cycles of ribbon graphs.
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A short overview of the "Topological recursion"

A short overview of the "Topological recursion"

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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Does the expansion of biofuels encroach on the forest?

Does the expansion of biofuels encroach on the forest?

The analysis of the biofuel-deforestation nexus is complex. The impact of biofuels on deforestation through land use change depends primarily on the type of crops used (Gao et al., 2012). The heterogeneity of the types of raw materials used in the production of bioethanol and biodiesel implies the existence of various transmission channels between biofuel production and deforestation. Not all crops are subject to the same type of land use changes and some are exploited on already agricultural or marginal land, especially in industrialized countries. Moreover, yield and price levels differ significantly by crop, which has an impact on production conditions and on the extent of land use change (Lapola et al., 2010). High crop yields allow an increase in the production of biofuels by an intensification of the exploitation of agricultural raw materials. In the case of indirect land use change, the productivity of the displaced agricultural activity comes into place. These changes depend partly on raw material market prices and on the demand elasticity (Lapola et al., 2010, Arima et al., 2011, Andrade de Sá et al., 2013). There are numerous studies at the global and national levels on the biofuel-deforestation nexus, but they are mainly based on simulations. Econometric studies are scant and mainly based on case studies at the subnational level. These studies allow easier access to accurate information about the types of raw materials used, their prices, yields and the share allocated to the production of biofuels (Gao et al., 2011).
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Diffusion at the Surface of Topological Insulators

Diffusion at the Surface of Topological Insulators

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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ON TOPOLOGICAL GENERICITY OF THE MODE-LOCKING PHENOMENON

ON TOPOLOGICAL GENERICITY OF THE MODE-LOCKING PHENOMENON

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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A topological definition of the Maslov bundle

A topological definition of the Maslov bundle

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,

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On traces of the Brandt-Eichler matrices

On traces of the Brandt-Eichler matrices

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):

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The Zeckendorf expansion of polynomial sequences

The Zeckendorf expansion of polynomial sequences

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,

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