# Gaëtan Borot

Since April 2013, I am researcher at the Max Planck Institute, Bonn. This old webpage will not be maintained anymore, the new webpage is here . Before, I have been a postdoc at University of Geneva, in the group of Stanislav Smirnov, and I spend my PhD years in theoretical physics at CEA Saclay, under the supervision of Bertrand Eynard. Earlier, I was a student at ENS Paris.

You can find here information on my research, a CV, upcoming events or slides of talks .

## Interests

I work at the interface of statistical physics and algebraic geometry. I started in random matrix theory, and I am trying to apply mathematical ideas coming from there to other fields, like enumerative geometry and topological strings, combinatorics of maps, integrable systems, and more recently knot theory. A common tool appearing in these topics is a recursion relation, called topological recursion and initially formalized by Eynard, Orantin and Chekhov. I am since then interested in Gromov-Witten theory and the BKMP proposition.

A very useful masterclass on the topic followed by a conference was organized in QGM, Aarhus in January 2013, and you can find here the videos.

With Eynard and Orantin, we have developed an abstract version of the topological recursion, which takes weaker assumptions than the initial one, and enlarges the set of possible applications:

Abstract loop equations, topological recursion, and applications , with B. Eynard and N. Orantin.
preprint (2013), math-ph/1303.5808

## Publications

### Enumerative geometry

In my first paper, we proved that an appropriate generating series for the number of simple coverings of the sphere by a Riemann surface of genus $g$, obeys the topological recursion of the Lambert curve $ye^{-y} = e^{-x}$.

A matrix model for simple Hurwitz numbers, and topological recursion, with B. Eynard, M. Mulase and B. Safnuk.
J. Geom. Phys. 61 (2011), math-ph/0906.1206

I am since then interested in Gromov-Witten theory and the BKMP conjecture.

### Combinatorics of maps

I study statistical physics models, like the $O(n)$ model or the Potts model , on random lattices of all topologies. A first problem is to enumerate (decorated) maps, i.e. how many inequivalent surfaces do can we obtain by gluing (with special rules) polygons along their edges ? The final goal is to understand the geometry of large random lattices , in relation with conformal field theory and SLE processes.

In the $O(n)$ model, maps carry self-avoiding loops with a weight $n$ for each (see the picture). In my second paper, we showed the enumeration problem in the trivalent $O(n)$ model is solved by a twisted version of the topological recursion.

Enumeration of maps with self-avoiding loops and the $O(n)$ model on random lattices of all topologies
with B. Eynard.
J. Stat. Mech. P01010 (2011), math-ph/0910.5896

I extended later this result to a larger class of $O(n)$ models. In a joint project with Jérémie Bouttier and Emmanuel Guitter from CEA Saclay, we study them by cutting along loops (these papers are written in a combinatorialist-friendly style). By duality between the $n^2$-Potts model and a loop model with $n$ colors, we could also study the Potts model on general random lattices with defects.

A recursive approach to the $O(n)$ model on random maps via nested loops, with J. Bouttier and E. Guitter.
J. Phys. A: Math. Theor. 45 045002 (2012), math-ph/1106.0153

More on the $O(n)$ model on random maps via nested loops: loops with bending energy, with J. Bouttier and E. Guitter.
J. Phys. A: Math. Theor. 45 275206 (2012), math-ph/1202.5521

Loop models on random maps via nested loops: case of domain symmetry breaking and application to the Potts model , with J. Bouttier and E. Guitter.
to appear in J. Phys. A, special issue (2012), math-ph/1207.4878

### Random matrix theory

I am interested in random matrix theory, especially in $\beta$ ensembles, i.e. probability measures on $\mathbb{R}^N$ of the form $\mathrm{d}\mu(\lambda_1,\ldots,\lambda_N) = \prod_{i = 1}^N \mathrm{d}\lambda_i\,g(\lambda_i)\,\prod_{1 \leq i \neq j \leq N} |\lambda_i - \lambda_j|^{\beta}$. The $\lambda_i$ can be thought as eigenvalues of certain random operators.

In particular, the $\beta$ Tracy-Widom distribution appears as the limit distribution of $\mathrm{max}_i\,\lambda_i$. Apart from $\beta = 1/2,1$ and $2$ where $\mathsf{TW}_{\beta}$ can be computed in relation with integrable systems and Painlevé equations (graph courtesy of J.M. Stéphan), few is known on $\mathsf{TW}_{\beta}$.

We showed how to compute (without a rigorous proof) the all order asymptotic expansion of the left and right tails of $\mathsf{TW}_{\beta}$.

Large deviations of the maximal eigenvalue of random matrices, with B. Eynard, S.N. Majumdar and C. Nadal.
J. Stat. Mech P11024 (2011), math-ph/1009.1945

Right tail expansion of Tracy-Widom beta laws, with C. Nadal.
Random Matrices: Theory Appl. (2012), math-ph/1111.2761

For $\beta = 1$ (hermitian random matrices), we proved that our result at the left tail matches the expression found by Tracy and Widom.

The asymptotic expansion of Tracy-Widom GUE law and symplectic invariants, with B. Eynard.
math-ph/1012.2752, preprint

With Alice Guionnet, we proved a technical result on $\beta$ ensembles, namely the existence of a $1/N$ expansion of the partition function under natural assumptions (the one-cut regime). Our result justifies one of the steps in the derivation of asymptotics of $\mathsf{TW}_{\beta}$

Asymptotic expansion of beta matrix models in the one-cut regime, with A. Guionnet.
math.PR/1107.1167, to appear in Commun. Math. Phys. (2012)

We extended this work to show the existence of the full asymptotic expansion in the (including oscillatory behavior when $N$ is large) in the multi-cut regime, thus justifying a heuristic derivation proposed earlier by Eynard. This allows to study the asymptotic expansion of orthogonal and skew-orthogonal polynomials by probabilistic methods, i.e. without relying on integrability.

Asymptotic expansion of beta matrix models in the multi-cut regime, with A. Guionnet.
math.PR/1303.1045, preprint.

We are now working on refinements of the results with weaker assumptions, which would be needed to study critical points of the model, and extension to multidimensional integrals with arbitrary pairwise interactions (instead of logarithmic interactions in $\beta$ ensembles). The latter would have applications e.g., in Chern-Simons theory on $3$-manifolds, or, in the physics of quantum integrable systems.

### Integrable systems and applications

I am studying the interplay between integrability and loop equations (also called Virasoro constraints). In a joint work with Bertrand Eynard, we conjectured that a certain solution of the loop equations, built from the topological recursion of an algebraic curve, and derivatives of theta function, is the tau function of an integrable system depending on a dispersive parameter.

Geometry of spectral curves and all order dispersive integrable system, with B. Eynard.
math-ph/1110.4936 , to appear in SIGMA (2013)

The theta functions represent non-perturbative effects. We conjectured that this formalism can be applied to compute the all-order asymptotics of the Jones polynomial of a hyperbolic knot. We checked it to first order for the figure-eight knot depicted here. This improves a conjecture of Dijkgraaf, Fuji, and Manabe (who considered only the "perturbative part" and found some mismatch) concerning the so-called "generalized volume conjecture". Following the conference

All order asymptotics of hyperbolic knot invariants from non-perturbative topological recursion of A-polynomials , with B. Eynard.
math-ph/1205.2261 , submitted.

We are preparing a paper going the other way round: starting with a rational Lax systems (=an integrable system), we shall give a result which, under some assumptions, allows to compute the all order WKB expansion from the topological recursion, and we illustrate it on examples. I am also collaborating with Boris Dubrovin, Tamara Grava and Stefano Romano (SISSA) to establish correspondences between the framework of Frobenius manifolds and integrable hierarchies of topological type, and the topological recursion.

### Statistical physics

Linear singular integral equations intervenes in many areas of statistical physics, often in relation with thermodynamical Bethe Ansatz. The techniques developed for the $O(n)$ model turned out to have applications to other problems in physics, like:

• Quantum entanglement. Collaboration with Céline Nadal at Rudolf Peierls Center for Theoretical Physics in Oxford.

Purity distribution for generalized random Bures mixed states, with C. Nadal.
J. Phys. A: Math. Theor. 45 (2012), cond-mat.stat-mech/1110.3838

• Combustion-related problems. Ongoing collaboration with Bruno Denet from IRPHE in Marseille, and Guy Joulin from Institut P' in Poitiers.

Resolvent methods for steady premixed flame shapes governed by the Zhdanov-Trubnikov equation, with B. Denet and G. Joulin.
J. Stat. Mech. (2012), P10023, physics.flu-dyn/1207.5416