# 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

*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)

*Asymptotic expansion of beta matrix models in the multi-cut regime*, with A. Guionnet.

math.PR/1303.1045, preprint.

### 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)

* All order asymptotics of hyperbolic knot invariants from non-perturbative topological recursion of A-polynomials *, with B. Eynard.

math-ph/1205.2261 , submitted.

### 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

Last modified: April 3rd 2013.