Institut Camille Jordan

Weingarten calculus addresses the problem of computing the integral of polynomial functions against the Haar measure over compact groups. In the case of classical series, it gives a rational fraction depending on the dimension. In a natural sense, it extends Gaussian (Wick) calculus. We will explain how these computations work, and focus on the case of tensor matrix integrals, whose study we initiated in a series of work in collaboration with Luca Lionni and Razvan Gurau.

Powerful computers and acquisition devices have made it possible to capture and store large real-world multidimensional data. For practical applications, analyzing and organizing these high dimensional arrays (formally called tensors) lead to the well-known curse of dimensionality. Thus, dimensionality reduction is frequently employed to transform a high-dimensional data set by projecting it into a lower dimensional space while retaining most of the information and underlying structure.  One of these techniques is Principal Component Analysis (PCA), which has made remarkable progress in a large number of areas thanks to its simplicity and adaptability.  

These last years, tools based on tensor contractions (trace invariants) have been developed by theoretical physicists where random tensors have emerged as a generalization of random matrices.  In this work, we investigate the algorithmic threshold of tensor PCA and some of its variants using the theoretical physics approach and we show that it leads to new insights and knowledge in tensor PCA.

Observables in a tensor model can be enumerated using colored graphs, which have a description in terms of permutation triples subject to an  equivalence relation generated by permutation products. A gauge-fixing of the equivalence relation relates the enumeration to bipartite ribbon graphs and Belyi maps between  surfaces. Fourier transformation on permutation group algebras relates the counting of observables to a sum of squares of Kronecker coefficients. In the recent paper https://arxiv.org/abs/2010.04054, we develop these observations to give a combinatoric construction of the Kronecker coefficient for a fixed triple of Young diagrams. The Kronecker coefficient counts vectors in a lattice of ribbon graphs, determined as null vectors of integer matrices.  The result motivates a discussion of  quantum mechanical systems and algorithms to determine non-vanishing Kronecker coefficients.  Using the link to Belyi maps, these quantum mechanical systems have an interpretation in terms of quantum membrane geometries interpolating between algebraic string worldsheets.

In this talk, I will explain how to build from a classical rational spectral curve (P(x,y)=0) a "quantum curve" P(x,\hbar\partial_x)\Psi=0 using the Chekhov-Eynard-Orantin topological recursion. The strategy is to regroup the correlation functions generated by the topological recursion into a matrix wave function \Psi and show that it satisfies a rational linear differential system with the same pole structure as the initial curve. In the hyper-elliptic case, we explain why this procedure is naturally connected with isomonodromic deformations and recovers the standard Lax pair formulations (Painlevé). The talk is mostly based on arxiv:1911.07739 with N. Orantin.

Bicolored maps with boundaries can be seen as a special, simple case of the Ising model on maps. However, they are yet to be enumerated explicitly. I will present in this talk a work in progress in collaboration with Jérémie Bouttier, where we used analytic-combinatorial tools for the specific case of planar bicolored maps with alternating boundaries. This yields an explicit rational parametrization in the case of m-angulations and m-constellations. In the special case of Eulerian triangulations, this rational parametrization was a key ingredient in proving in https://arxiv.org/abs/1912.13434 that this family of maps admits the Brownian map as a scaling limit.

After https://arxiv.org/abs/2007.04603.

Partly after https://arxiv.org/abs/1808.09434.

The Hermitian 1-matrix model has been known for 30 years to satisfy the
KP hierarchy. It is an infinite set of partial differential equations,
here with respect to the coupling constants of the model. At least three
proofs of this statement are known. Other systems in two-dimensional
enumerative geometry such as, famously, intersection numbers of the
moduli space of Riemann surfaces, satisfy the KP hierarchy. In this
talk, I will introduce the KP hierarchy from the point of view of the
Sato Grassmannian as an orbit of an action of GL(\infty). This will then
give us the opportunity to present the proof of Kazarian and Zograf that
the generating function of bipartite maps satisfies the KP hierarchy. I
will then focus on constellations, which are generalizations of
bipartite maps, and are also the dual to the jackets of the bipartite
edge-colored graphs of unitary-invariant tensor models. Constellations
are known to also satisfy the KP hierarchy but only a proof via
algebraic combinatorics is known. I will report some progress towards
understanding their integrable properties using
Tutte/Schwinger-Dyson--like equations. Only this last part is an
original contribution.

after https://arxiv.org/abs/1712.05670 and https://arxiv.org/abs/1910.13261.

after https://arxiv.org/abs/1910.13982.

After https://arxiv.org/abs/1912.05314.

After https://arxiv.org/abs/2004.07894.

It is known that the Kontsevich model is equivalent to the hermitian 1-matrix model by choosing a special relation between the moments of the external matrix (Kontsevich times) and the parameters of the potential of the hermitian 1-matrix model. This equivalence can easily be proven by topological recursion.
We will present the quartic analogue of the Kontsevich model with the action S=Tr(E \phi^2+\frac{\lambda}{4}\phi^4), where E is the external matrix and \lambda a real number. The Dyson-Schwinger equations (loop equations) will be shown, as well as some results for the first correlation functions. From these results the genus zero spectral curve of the model is identified, wich has a the symmetry x(z)=-y(-z). Furthermore, it turns out that the results and loop equations of the quartic model are structurally the same as from the hermitian 2-matrix model with a genus zero assumption for the spectral curve. The hermitian 2-matrix model satisfies a more general topological expansion.

This talk is partially based on https://arxiv.org/abs/1906.04600.

After https://arxiv.org/abs/2002.07652.

After https://arxiv.org/abs/1908.02259.

After https://arxiv.org/abs/2003.02100.

After https://arxiv.org/abs/2004.02660.