WKB solutions of difference equations and reconstruction by the topological recursion

The purpose of this article is to analyze the connection between Eynard-Orantin topological recursion and formal WKB solutions of a $\hbar$-difference equation: $\Psi(x+\hbar)=\left(e^{\hbar\frac{d}{dx}}\right) \Psi(x)=L(x;\hbar)\Psi(x)$ with $L(x;\hbar)\in GL_2( (\mathbb{C}(x))[\hbar])$. In particular, we extend the notion of determinantal formulas and topological type property proposed for formal WKB solutions of $\hbar$-differential systems to this setting. We apply our results to a specific $\hbar$-difference system associated to the quantum curve of the Gromov-Witten invariants of $\mathbb{P}^1$ for which we are able to prove that the correlation functions are reconstructed from the Eynard-Orantin differentials computed from the topological recursion applied to the spectral curve $y=\cosh^{-1}\frac{x}{2}$. Finally, identifying the large $x$ expansion of the correlation functions, proves a recent conjecture made by B. Dubrovin and D. Yang regarding a new generating series for Gromov-Witten invariants of $\mathbb{P}^1$.Comment: 41 pages, 2 figures, published version in Nonlinearit

Rectangular Young tableaux and the Jacobi ensemble

It has been shown by Pittel and Romik that the random surface associated with a large rectangular Young tableau converges to a deterministic limit. We study the fluctuations from this limit along the edges of the rectangle. We show that in the corner, these fluctuations are gaussian wheras, away from the corner and when the rectangle is a square, the fluctuations are given by the Tracy-Widom distribution. Our method is based on a connection with the Jacobi ensemble

Elements of proof for conjectures of Witte and Forrester about the combinatorial structure of Gaussian Beta Ensembles

The purpose of the article is to provide partial proofs for two conjectures given by Witte and Forrester in "Moments of the Gaussian $\beta$ Ensembles and the large $N$ expansion of the densities" with the use of the topological recursion adapted for general $\beta$ Gaussian case. In particular, the paper uses a version at coinciding points that provides a simple proof for some of the coefficients involved in the conjecture. Additionally, we propose a generalized version of the conjectures for all correlation functions evaluated at coinciding points.Comment: 18 pages, version accepted in JHEP, minor mistakes correcte

On the sub-Gaussianity of the Beta and Dirichlet distributions

We obtain the optimal proxy variance for the sub-Gaussianity of Beta distribution, thus proving upper bounds recently conjectured by Elder (2016). We provide different proof techniques for the symmetrical (around its mean) case and the non-symmetrical case. The technique in the latter case relies on studying the ordinary differential equation satisfied by the Beta moment-generating function known as the confluent hypergeometric function. As a consequence, we derive the optimal proxy variance for the Dirichlet distribution, which is apparently a novel result. We also provide a new proof of the optimal proxy variance for the Bernoulli distribution, and discuss in this context the proxy variance relation to log-Sobolev inequalities and transport inequalities.Comment: 13 pages, 2 figure

On the concentration of measure phenomenon for stable and related random vectors

Concentration of measure is studied, and obtained, for stable and related random vectors.Comment: Published by the Institute of Mathematical Statistics (http://www.imstat.org) in the Annals of Probability (http://www.imstat.org/aop/) at http://dx.doi.org/10.1214/00911790400000028

Modeling Location Choice of Secondary Activities with a Social Network of Cooperative Agents

Activity-based models in transportation science focus on the description of human trips and activities. Modeling the spatial decision for so-called secondary activities is addressed in this paper. Given both home and work locations, where do individuals perform activities such as shopping and leisure? Simulation of these decisions using random utility models requires a full enumeration of possible outcomes. For large data sets, it becomes computationally unfeasible because of the combinatorial complexity. To overcome that limitation, a model is proposed in which agents have limited, accurate information about a small subset of the overall spatial environment. Agents are interconnected by a social network through which they can exchange information. This approach has several advantages compared with the explicit simulation of a standard random utility model: (a) it computes plausible choice sets in reasonable computing times, (b) it can be extended easily to integrate further empirical evidence about travel behavior, and (c) it provides a useful framework to study the propagation of any newly available information. This paper emphasizes the computational efficiency of the approach for real-world examples

Loop equations from differential systems

To any differential system $d\Psi=\Phi\Psi$ where $\Psi$ belongs to a Lie group (a fiber of a principal bundle) and $\Phi$ is a Lie algebra $\mathfrak g$ valued 1-form on a Riemann surface $\Sigma$, is associated an infinite sequence of "correlators" $W_n$ that are symmetric $n$-forms on $\Sigma^n$. The goal of this article is to prove that these correlators always satisfy "loop equations", the same equations satisfied by correlation functions in random matrix models, or the same equations as Virasoro or W-algebra constraints in CFT.Comment: 20 page