Higher Order Analogues of Tracy-Widom Distributions via the Lax Method

We study the distribution of the largest eigenvalue in formal Hermitian one-matrix models at multicriticality, where the spectral density acquires an extra number of k-1 zeros at the edge. The distributions are directly expressed through the norms of orthogonal polynomials on a semi-infinite interval, as an alternative to using Fredholm determinants. They satisfy non-linear recurrence relations which we show form a Lax pair, making contact to the string literature in the early 1990's. The technique of pseudo-differential operators allows us to give compact expressions for the logarithm of the gap probability in terms of the Painleve XXXIV hierarchy. These are the higher order analogues of the Tracy-Widom distribution which has k=1. Using known Backlund transformations we show how to simplify earlier equivalent results that are derived from Fredholm determinant theory, valid for even k in terms of the Painleve II hierarchy.Comment: 24 pages. Improved discussion of Backlund transformations, in addition to other minor improvements in text. Typos corrected. Matches published versio

Right tail expansion of Tracy-Widom beta laws

Using loop equations, we compute the large deviation function of the maximum eigenvalue to the right of the spectrum in the Gaussian beta matrix ensembles, to all orders in 1/N. We then give a physical derivation of the all order asymptotic expansion of the right tail Tracy-Widom beta laws, for all positive beta, by studying the double scaling limit.Comment: 23 page

A recursive approach to the O(n) model on random maps via nested loops

We consider the O(n) loop model on tetravalent maps and show how to rephrase it into a model of bipartite maps without loops. This follows from a combinatorial decomposition that consists in cutting the O(n) model configurations along their loops so that each elementary piece is a map that may have arbitrary even face degrees. In the induced statistics, these maps are drawn according to a Boltzmann distribution whose parameters (the face weights) are determined by a fixed point condition. In particular, we show that the dense and dilute critical points of the O(n) model correspond to bipartite maps with large faces (i.e. whose degree distribution has a fat tail). The re-expression of the fixed point condition in terms of linear integral equations allows us to explore the phase diagram of the model. In particular, we determine this phase diagram exactly for the simplest version of the model where the loops are "rigid". Several generalizations of the model are discussed.Comment: 47 pages, 13 figures, final version (minor changes with v2 after proof corrections

Brunel-Dominated Proton Acceleration with a Few-Cycle Laser Pulse

International audienceExperimental measurements of backward accelerated protons are presented. The beam is produced when an ultrashort (5 fs) laser pulse, delivered by a kHz laser system, with a high temporal contrast (10 8), interacts with a thick solid target. Under these conditions, proton cutoff energy dependence with laser parameters, such as pulse energy, polarization (from p to s), and pulse duration (from 5 to 500 fs), is studied. Theoretical model and two-dimensional particle-in-cell simulations, in good agreement with a large set of experimental results, indicate that proton acceleration is directly driven by Brunel electrons, in contrast to conventional target normal sheath acceleration that relies on electron thermal pressure

Spectral density asymptotics for Gaussian and Laguerre β\beta-ensembles in the exponentially small region

The first two terms in the large $N$ asymptotic expansion of the $\beta$ moment of the characteristic polynomial for the Gaussian and Laguerre $\beta$-ensembles are calculated. This is used to compute the asymptotic expansion of the spectral density in these ensembles, in the exponentially small region outside the leading support, up to terms $o(1)$ . The leading form of the right tail of the distribution of the largest eigenvalue is given by the density in this regime. It is demonstrated that there is a scaling from this, to the right tail asymptotics for the distribution of the largest eigenvalue at the soft edge.Comment: 19 page

Resolvent methods for steady premixed flame shapes governed by the Zhdanov-Trubnikov equation

Using pole decompositions as starting points, the one parameter (-1 =< c < 1) nonlocal and nonlinear Zhdanov-Trubnikov (ZT) equation for the steady shapes of premixed gaseous flames is studied in the large-wrinkle limit. The singular integral equations for pole densities are closely related to those satisfied by the spectral density in the O(n) matrix model, with n = -2(1 + c)/(1 - c). They can be solved via the introduction of complex resolvents and the use of complex analysis. We retrieve results obtained recently for -1 =< c =< 0, and we explain and cure their pathologies when they are continued naively to 0 < c < 1. Moreover, for any -1 =< c < 1, we derive closed-form expressions for the shapes of steady isolated flame crests, and then bicoalesced periodic fronts. These theoretical results fully agree with numerical resolutions. Open problems are evoked.Comment: v2: 29 pages, 6 figures, some typos correcte

Enumeration of maps with self avoiding loops and the O(n) model on random lattices of all topologies

We compute the generating functions of a O(n) model (loop gas model) on a random lattice of any topology. On the disc and the cylinder, they were already known, and here we compute all the other topologies. We find that the generating functions (and the correlation functions of the lattice) obey the topological recursion, as usual in matrix models, i.e they are given by the symplectic invariants of their spectral curve.Comment: pdflatex, 89 pages, 12 labelled figures (15 figures at all), minor correction

From Hurwitz numbers to Kontsevich-Witten tau-function: a connection by Virasoro operators

In this letter,we present our conjecture on the connection between the Kontsevich--Witten and the Hurwitz tau-functions. The conjectural formula connects these two tau-functions by means of the $GL(\infty)$ group element. An important feature of this group element is its simplicity: this is a group element of the Virasoro subalgebra of $gl(\infty)$. If proved, this conjecture would allow to derive the Virasoro constraints for the Hurwitz tau-function, which remain unknown in spite of existence of several matrix model representations, as well as to give an integrable operator description of the Kontsevich--Witten tau-function.Comment: 13 page

A simple derivation of the Tracy-Widom distribution of the maximal eigenvalue of a Gaussian unitary random matrix

In this paper, we first briefly review some recent results on the distribution of the maximal eigenvalue of a $(N\times N)$ random matrix drawn from Gaussian ensembles. Next we focus on the Gaussian Unitary Ensemble (GUE) and by suitably adapting a method of orthogonal polynomials developed by Gross and Matytsin in the context of Yang-Mills theory in two dimensions, we provide a rather simple derivation of the Tracy-Widom law for GUE. Our derivation is based on the elementary asymptotic scaling analysis of a pair of coupled nonlinear recursion relations. As an added bonus, this method also allows us to compute the precise subleading terms describing the right large deviation tail of the maximal eigenvalue distribution. In the Yang-Mills language, these subleading terms correspond to non-perturbative (in $1/N$ expansion) corrections to the two-dimensional partition function in the so called `weak' coupling regime.Comment: 2 figure

Challenges of beta-deformation

A brief review of problems, arising in the study of the beta-deformation, also known as "refinement", which appears as a central difficult element in a number of related modern subjects: beta \neq 1 is responsible for deviation from free fermions in 2d conformal theories, from symmetric omega-backgrounds with epsilon_2 = - epsilon_1 in instanton sums in 4d SYM theories, from eigenvalue matrix models to beta-ensembles, from HOMFLY to super-polynomials in Chern-Simons theory, from quantum groups to elliptic and hyperbolic algebras etc. The main attention is paid to the context of AGT relation and its possible generalizations.Comment: 20 page