Related papers: Universality and asymptotics of graph counting pro…
In this work we propose a new method for investigating connection problems for the class of nonlinear second-order differential equations known as the Painlev{\'e} equations. Such problems can be characterized by the question as to how the…
The generating function which counts partitions with the Plancherel measure (and its q-deformed version), can be rewritten as a matrix integral, which allows to compute its asymptotic expansion to all orders. There are applications in…
For equation P$_I^2$, the second member in the P$_I$ hierarchy, we prove existence of various degenerate solutions depending on the complex parameter $t$ and evaluate the asymptotics in the complex $x$ plane for $|x|\to\infty$ and…
Rational solutions of the inhomogeneous Painleve-II equation and of a related coupled Painleve-II system have recently arisen in studies of fluid vortices and of the sine-Gordon equation. For the sine-Gordon application in particular it is…
The computation of observables in general interacting theories, be them quantum mechanical, field, gauge or string theories, is a non-trivial problem which in many cases can only be addressed by resorting to perturbative methods. In most…
For analytic nonlinear systems of ordinary differential equations, under some non-degeneracy and integrability conditions we prove that the formal exponential series solutions (trans-series) at an irregular singularity of rank one are Borel…
We consider a class of $n^{\text{th}}$-order linear ordinary differential equations with a large parameter $u$. Analytic solutions of these equations can be described by (divergent) formal series in descending powers of $u$. We demonstrate…
We investigate geometric aspects of co-equational parametric resurgence, by studying physical problems whose formal asymptotic solutions give rise to Borel transforms lying on an algebraic curve. This perspective allows us to elucidate…
We analyze the one parameter family of tronqu\'ee solutions of the Painlev\'e equation \P1 in the pole-free sectors together with the region of the first array of poles. We find a convergent expansion for these solutions, containing one…
We address the enumeration of Eulerian orientations of 4-valent planar maps according to three parameters: the number of vertices, the number of alternating vertices (having in/out/in/out incident edges), and the number of clockwise…
We provide a general framework for proving asymptotic equidistribution, convexity, and log concavity of coefficients of generating functions on arithmetic progressions. Our central tool is a variant of Wright's Circle Method proved by two…
In a recent article J. Phys. Compl. 4 (2023) 035005, Kawamoto evoked statistical physics methods for the problem of counting graphs with a prescribed degree sequence. This treatment involved truncating a particular Taylor expansion at the…
We consider the ideal orientation problem in planar graphs. In this problem, we are given an undirected graph $G$ with positive edge lengths and $k$ pairs of distinct vertices $(s_1, t_1), \dots, (s_k, t_k)$ called terminals, and we want to…
We use the method of higher order linearization to study an inverse boundary value problem for the minimal surface equation on a Riemannian manifold $(\mathbb{R}^n,g)$, where the metric $g$ is conformally Euclidean. In particular we show…
This article concerns the computational complexity of a fundamental problem in number theory: counting points on curves and surfaces over finite fields. There is no subexponential-time algorithm known and it is unclear if it can be…
We analyze truncated series generated as divergent formal solutions of non-linear ordinary differential equations. Motivating the study is a specific non-linear, first-order differential equation, which is the basis of the resurgent…
We elucidate the relation between Painlev\'e equations and four-dimensional rank one ${\cal N= 2}$ theories by identifying the connection associated to Painlev\'e isomonodromic problems with the oper limit of the flat connection of the…
Let $p$ be a prime number and consider a $p$-automatic sequence ${\bf u}=(u_{n})_{n\in\N}$ and its generating function $U(X)=\sum_{n=0}^{\infty}u_{n}X^{n}\in\mathbb{F}_{p}[[X]]$. Moreover, let us suppose that $u_{0}=0$ and $u_{1}\neq 0$ and…
We consider the planar orthogonal polynomial $p_{n}(z)$ with respect to the measure supported on the whole complex plane $${\rm e}^{-N|z|^2} \prod_{j=1}^\nu |z-a_j|^{2c_j}\,{\rm d} A(z)$$ where ${\rm d} A$ is the Lebesgue measure of the…
A unicellular map is the embedding of a connected graph in a surface in such a way that the complement of the graph is a topological disk. In this paper we present a bijective link between unicellular maps on a non-orientable surface and…