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Register a new userThe sine-law gap probability, Painleve 5, and asymptotic expansion by the topological recursion
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The goal of this article is to rederive the connection between the Painleve $5$ integrable system and the universal eigenvalues correlation functions of double-scaled hermitian matrix models, through the topological recursion method. More specifically we prove, textbf{to all orders}, that the WKB asymptotic expansions of the $tau$-function as well as of determinantal formulas arising from the Painleve $5$ Lax pair are identical to the large $N$ double scaling asymptotic expansions of the partition function and correlation functions of any hermitian matrix model around a regular point in the bulk. In other words, we rederive the sine-law universal bulk asymptotic of large random matrices and provide an alternative perturbative proof of universality in the bulk with only algebraic methods. Eventually we exhibit the first orders of the series expansion up to $O(N^{-5})$.
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Starting from a $dtimes d$ rational Lax pair system of the form $hbar partial_x Psi= LPsi$ and $hbar partial_t Psi=RPsi$ we prove that, under certain assumptions (genus $0$ spectral curve and additional conditions on $R$ and $L$), the system satisfies the topological type property. A consequence is that the formal $hbar$-WKB expansion of its determinantal correlators, satisfy the topological recursion. This applies in particular to all $(p,q)$ minimal models reductions of the KP hierarchy, or to the six Painleve systems.
The short-distance expansion of the tau function of the radial sine-Gordon/Painleve III equation is given by a convergent series which involves irregular $c=1$ conformal blocks and possesses certain periodicity properties with respect to monodromy data. The long-distance irregular expansion exhibits a similar periodicity with respect to a different pair of coordinates on the monodromy manifold. This observation is used to conjecture an exact expression for the connection constant providing relative normalization of the two series. Up to an elementary prefactor, it is given by the generating function of the canonical transformation between the two sets of coordinates.
This review is an extended version of the Seoul ICM 2014 proceedings.It is a short overview of the topological recursion, a relation appearing in the asymptotic expansion of many integrable systems and in enumerative problems. We recall how computing large size asymptotics in random matrices, has allowed to discover some fascinating and ubiquitous geometric invariants. Specializations of this method recover many classical invariants, like Gromov--Witten invariants, or knot polynomials (Jones, HOMFLY,...). In this short review, we give some examples, give definitions, and review some properties and applications of the formalism.
The purpose of this article is to analyze the connection between Eynard-Orantin topological recursion and formal WKB solutions of a $hbar$-difference equation: $Psi(x+hbar)=left(e^{hbarfrac{d}{dx}}right) Psi(x)=L(x;hbar)Psi(x)$ 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 $mathbb{P}^1$.
We show that for a rather generic set of regular spectral curves, the Topological-Recursion invariants F_g grow at most like $O((beta g)! r^{-g}) $ with some $r>0$ and $betaleq 5$.
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