Formal multidimensional integrals, stuffed maps, and topological recursion

Borot, Gaëtan

doi:10.4171/aihpd/7

Cited by 36 publications

(94 citation statements)

References 15 publications

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“…In [16], we discovered that to study more general Hermitian matrix models, we must introduce the blobs. We also notice a similarity between blobs and nonperturbative effects in matrix models, and this similarity demands further investigation.…”

Section: Topological Recursionmentioning

confidence: 99%

Blobbed topological recursion

Borot

2015

Theor Math Phys

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We describe the formalism and the properties of the blobbed topological recursion, which provides the general solution of a set of abstract loop equations. This procedure extends the topological recursion by introducing extra terms (blobs) in the initial conditions for each multidifferential ωg,n. We apply this formalism to topological expansion of formal Hermitian matrix models (the blobs are necessary to include arbitrary interactions) and pose some open questions.This paper is a short, preliminary version of the extended electronic article [1]. Axiomatic definitionFunctional equations are encountered in the enumerative geometry of surfaces: Virasoro constraints (in matrix models and intersection theory over M g,n ), Tutte's equations (in map enumeration), cut-andjoin equations (in covering enumerations), etc. They have very similar natures, which has to do with the recursive construction of orientable surfaces. We introduced abstract loop equations in [2] as an attempt to write a universal set of equations, to which the abovementioned functional equations can be reduced, maybe after some problem-specific technical steps. Abstract loop equations.Let U i , i ∈ I, be a finite collection of open sets that are complex one-dimensional manifolds with a distinguished point α i ∈ U i . We can take an arbitrary small neighborhood of α i as U i because our secret purpose is to speak of germs of functions and differential forms at α i . We assume that U i comes with a covering {x : U i → V i ⊆ C} that has a ramification point at α i . We also assume that all ramification points are simple, i.e., α i is a simple zero of dx, although the theory can be formulated for higher-order ramifications following [3], [4]. We then have a deck transformation ι : U i → U i , i.e., a holomorphic map such that x(ι(z)) = x(z) but ι = id (see Fig.

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Section: Topological Recursionmentioning

confidence: 99%

Blobbed topological recursion

Borot

2015

Theor Math Phys

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“…One notices that expanding the logarithm of (3.11) the matrix model one obtains looks like the matrix models introduced in [6] for specific choices of the values of the formal variables. The models introduced therein are studied because of their connection to LMO invariants of 3-manifolds (see [7]).…”

Section: T 4 Tensor Models and Intermediate Field Representationmentioning

confidence: 99%

“…For instance an orthogonal polynomial is provided by P N,t 4 (x) = det(x − M ) . Following [12], changing variables to eigenvalues leads to: 6) where dµ(x) = exp(−N V (x))dx. Then write:…”

Section: Jhep08(2015)129mentioning

confidence: 99%

A Givental-like formula and bilinear identities for tensor models

Dartois

2015

J. High Energ. Phys.

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“…For general r, Borot [23] has shown that the formal asymptotic expansion of the partition function subordinate to multi-particle potentials is captured by a generalisation of the topological recursion. The existence of the all-order asymptotic expansion was established by the authors in [30] under certain regularity assumptions on the multi-particle interactions.…”

Section: Generalisationsmentioning

confidence: 99%

“…For β = 2, with the same precise meaning that was discussed in Section 1.1.2, the N-fold integral: [23], and encompasses a large class of statistical physics models on two-dimensional random lattices. The simplest ones occur for r = 2, and the only other interesting cases we are aware of have r = ∞.…”

Section: Statistical Physics In Two-dimensional Random Latticesmentioning

confidence: 99%

Asymptotic Expansion of a Partition Function Related to the Sinh-model

Borot¹,

Guionnet²,

Kozłowski³

2016

Mathematical Physics Studies

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This paper develops a method to carry out the large-N asymptotic analysis of a class of Ndimensional integrals arising in the context of the so-called quantum separation of variables method. We push further ideas developed in the context of random matrices of size N, but in the present problem, two scales 1/N α and 1/N naturally occur. In our case, the equilibrium measure is N α -dependent and characterised by means of the solution to a 2×2 Riemann-Hilbert problem, whose large-N behaviour is analysed in detail. Combining these results with techniques of concentration of measures and an asymptotic analysis of the Schwinger-Dyson equations at the distributional level, we obtain the large-N behaviour of the free energy explicitly up to o(1). The use of distributional Schwinger-Dyson is a novelty that allows us treating sufficiently differentiable interactions and the mixing of scales 1/N α and 1/N, thus waiving the analyticity assumptions often used in random matrix theory.1 gborot@mpim-bonn.mpg.de 2 guionnet@math.mit.edu 3 karol.kozlowski@ens-lyon.fr 2 An opening discussionThe present work develops techniques enabling one to carry out the large-N asymptotic analysis of a class of multiple integrals that arise as representations for the correlation functions in quantum integrable models solvable by the quantum separation of variables. We shall refer to the general class of such integrals as the sinh model: [9,59,60,82,106,105,139,146]. The expressions obtained there are either directly of the form given above or are amenable to this form (with, possibly, a change of the integration contour from R N to C N , with C a curve in C) upon elementary manipulations. Furthermore, a degeneration of z N [W] arises as a multiple integral representation for the partition function of the six-vertex model subject to domain wall boundary conditions [93]. In the context of quantum integrable systems, the number N of integrals defining z N is related to the number of sites in a model (as, e.g. in the case of the compact or non-compact XXZ chains or the lattice regularisations of the Sinh or Sine-Gordon models) or to the number of particles (as, e.g. in the case of the quantum Toda chain). From the point of view of applications, one is mainly interested in the thermodynamic limit of the model, which is attained by sending N to +∞. For instance, in the case of an integrable lattice discretisation of some quantum field theory, one obtains in this way an exact and non-perturbative description of a quantum field theory in 1 + 1 dimensions and in finite volume. This limit, at the level of z N [W], translates itself in the need to extract the large N-asymptotic expansion of ln z N [W] up to o(1). It is, in fact, the constant term in the expansion of ln zwith W ′ some deformation of W that gives rise to the correlation functions of the underlying quantum field theory in finite volume. These applications to physics constitute the first motivation for our analysis. From the purely mathematical side, the motivation of our works stems ...

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Formal multidimensional integrals, stuffed maps, and topological recursion

Cited by 36 publications

References 15 publications

Blobbed topological recursion

Blobbed topological recursion

A Givental-like formula and bilinear identities for tensor models

Asymptotic Expansion of a Partition Function Related to the Sinh-model

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