arXiv subjects
Publications and source records attributed to Marco Bertola.
We derive a particular solution of the extended $r$-reduced KP hierarchy, which is specified by a generalized string equation. The work is a generalization to arbitrary $r\geq 2$ of Buryak's recent results of a solution to the extended open KdV hierarchy which corresponds to $r=2$.
We study the asymptotics of singular values and singular functions of a Finite Hilbert transform (FHT), which is defined on several intervals. Transforms of this kind arise in the study of the interior problem of tomography. We suggest a novel approach based on the technique of the matrix Riemann-Hilbert problem and the steepest descent method of Deift-Zhou. We obtain a family of matrix RHPs depending on the spectral parameter $λ$ and show that the singular values of the FHT coincide with the values of $λ$ for which the RHP is not solvable. Expressing the leading order solution as $λ\to 0$ of the RHP in terms of the Riemann Theta functions, we prove that the asymptotics of the singular values can be obtained by studying the intersections of the locus of zeroes of a certain Theta function with a straight line. This line can be calculated explicitly, and it depends on the geometry of the intervals that define the FHT. The leading order asymptotics of the singular functions and singular values are explicitly expressed in terms of the Riemann Theta functions and of the period matrix of the corresponding normalized differentials, respectively. We also obtain the error estimates for our asymptotic results.
In this paper we continue investigation of the interior problem of tomography that was started in \cite{BKT2}. As is known, solving the interior problem {with prior data specified on a finite collection of intervals $I_i$} is equivalent to analytic continuation of a function from $I_i$ to an open set ${\bf J}$. In the paper we prove that this analytic continuation can be obtained with the help of a simple explicit formula, which involves summation of a series. Our second result is that the operator of analytic continuation is not stable for any pair of Sobolev spaces regardless of how close the set ${\bf J}$ is to $I_i$. Our main tool is the singular value decomposition of the operator $\mathcal H^{-1}_e$ that arises when the interior problem is reduced to a problem of inverting the Hilbert transform from incomplete data. The asymptotics of the singular values and singular functions of $\mathcal H^{-1}_e$, the latter being valid uniformly on compact subsets {of the interior of $I_i$}, was obtained in \cite{BKT2}. {Using these asymptotics we can accurately measure the degree of ill-posedness of the analytic continuation as a function of the target interval ${\bf J}$.} Our {last} result is the convergence of the asymptotic approximation of the singular functions {in the $L^2(I_i)$ sense}.
In this paper we describe a general method to derive formulas relating the gap probability of some classical determinantal random point process (Airy, Pearcey and Hermite) with the gap probability of the processes related to the same kernels with "wanderers", "inliers" and "outliers". In this way, we generalize the Painlevé-like formula found by Baik for the Baik-Ben Arous-Péché distribution to many different cases, both in the one and multi-time case. The method is not ad-hoc and relies upon the notion of discrete Schlesinger transformations for Riemann-Hilbert problems.
In the present paper we derive a new Hankel determinant representation for the square of the Vorob'ev-Yablonski polynomial $\mathcal{Q}_n(x),x\in\mathbb{C}$. These polynomials are the major ingredients in the construction of rational solutions to the second Painlevé equation $u_{xx}=xu+2u^3+α$. As an application of the new identity, we study the zero distribution of $\mathcal{Q}_n(x)$ as $n\rightarrow\infty$ by asymptotically analyzing a certain collection of (pseudo) orthogonal polynomials connected to the aforementioned Hankel determinant. Our approach reproduces recently obtained results in the same context by Buckingham and Miller \cite{BM}, which used the Jimbo-Miwa Lax representation of PII equation and the asymptotical analysis thereof.
We consider the orthogonal polynomials $\{P_{n}(z)\}$ with respect to the measure $|z-a|^{2N c} {\rm e}^{-N |z|^2} \,{\rm d} A(z)$ over the whole complex plane. We obtain the strong asymptotic of the orthogonal polynomials in the complex plane and the location of their zeros in a scaling limit where $n$ grows to infinity with $N$. The asymptotics are described in terms of three (probability) measures associated with the problem. The first measure is the limit of the counting measure of zeros of the polynomials, which is captured by the $g$-function much in the spirit of ordinary orthogonal polynomials on the real line. The second measure is the equilibrium measure that minimizes a certain logarithmic potential energy, supported on a region $K$ of the complex plane. The third measure is the harmonic measure of $K^c$ with a pole at $\infty$. This appears as the limit of the probability measure given (up to the normalization constant) by the squared modulus of the $n$-th orthogonal polynomial times the orthogonality measure, i.e. $|P_n(z)|^2 |z-a|^{2N c} {\rm e}^{-N |z|^2} \,{\rm d} A(z)$. The compact region $K$ that is the support of the second measure undergoes a topological transition under the variation of the parameter $t=n/N$; in a double scaling limit near the critical point given by $t_c=a(a+2\sqrt c)$ we observe the Hastings-McLeod solution to Painlevé\ II in the asymptotics of the orthogonal polynomials.
We apply the nonlinear steepest descent method to a class of 3x3 Riemann-Hilbert problems introduced in connection with the Cauchy two-matrix random model. The general case of two equilibrium measures supported on an arbitrary number of intervals is considered. In this case, we solve the Riemann-Hilbert problem for the outer parametrix in terms of sections of a spinorial line bundle on a three-sheeted Riemann surface of arbitrary genus and establish strong asymptotic results for the Cauchy biorthogonal polynomials.
The semiclassical (zero-dispersion) limit of the one-dimensional focusing Nonlinear Schroedinger equation (NLS) with decaying potentials is studied in a full scaling neighborhood D of the point of gradient catastrophe (x_0,t_0). This neighborhood contains the region of modulated plane wave (with rapid phase oscillations), as well as the region of fast amplitude oscillations (spikes). In this paper we establish the following universal behaviors of the NLS solutions near the point of gradient catastrophe: i) each spike has the height 3|q_0(x_0,t_0,epsilon)| and uniform shape of the rational breather solution to the NLS, scaled to the size O(epsilon); ii) the location of the spikes are determined by the poles of the tritronquee solution of the Painleve I (P1) equation through an explicit diffeomorphism between D and a region into the Painleve plane; iii) if (x,t) belongs to D but lies away from the spikes, the asymptotics of the NLS solution q(x,t,epsilon) is given by the plane wave approximation q_0(x,t,epsilon), with the correction term being expressed in terms of the tritronquee solution of P1. The latter result confirms the conjecture of Dubrovin, Grava and Klein about the form of the leading order correction in terms of the tritronquee solution in the non-oscillatory region around (x_0,t_0). We conjecture that the P1 hierarchy occurs at higher degenerate catastrophe points and that the amplitudes of the spikes are odd multiples of the amplitude at the corresponding catastrophe point. Our technique is based on the nonlinear steepest descent method for matrix Riemann-Hilbert Problems and discrete Schlesinger isomonodromic transformations.
Using a simple operator-norm estimate we show that the solution to the second Painlevé equation within the Ablowitz-Segur family is pole-free in a well defined region of the complex plane of the independent variable. The result is illustrated with several numerical examples.
We study the asymptotics of recurrence coefficients for monic orthogonal polynomials p_n(z) with the quartic exponential weight exp [-N (1/2 z^2 + t/4 z^4)], where t is complex. Our goals are: A) to describe the regions of different asymptotic behaviour (different genera) globally in t; B) to identify all the critical points, and; C) to study in details the asymptotics in a full neighborhood near of critical points (double scaling limit), including at and near the poles of Painleve' I solutions y(v) that are known to provide the leading correction term in this limit. Our results are: A) We found global (in t) asymptotic of recurrence coefficients and of "square-norms" for the orthogonal polynomials for different configurations of the contours of integration. Special code was developed to analyze all possible cases. B) In addition to the known critical point t_0=- 1/ 12, we found new critical points t_1=1/15 and t_2=1/4. C) We derived the leading order behavior of the recurrence coefficients (together with the error estimates) at and around the poles of y(v) near the critical points t_0,t_1 in what we called the triple scaling limit. We proved that the recurrence coefficients have unbounded "spikes" near the poles of y(v) and calculated the "universal" shape of these spikes for different cases. The nonlinear steepest descent method for Riemann-Hilbert Problem (RHP) is the main technique used in the paper. We note that the RHP near the critical points is very similar to the RHP describing the semiclassical limit of the focusing NLS near the point of gradient catastrophe that the authors solved previously.
Random Hermitian matrices are used to model complex systems without time-reversal invariance. Adding an external source to the model can have the effect of shifting some of the matrix eigenvalues, which corresponds to shifting some of the energy levels of the physical system. We consider the case when the $n\times n$ external source matrix has two distinct real eigenvalues: $a$ with multiplicity $r$ and zero with multiplicity $n-r$. For a Gaussian potential, it was shown by Péché \cite{Peche:2006} that when $r$ is fixed or grows sufficiently slowly with $n$ (a small-rank source), $r$ eigenvalues are expected to exit the main bulk for $|a|$ large enough. Furthermore, at the critical value of $a$ when the outliers are at the edge of a band, the eigenvalues at the edge are described by the $r$-Airy kernel. We establish the universality of the $r$-Airy kernel for a general class of analytic potentials for $r=\mathcal{O}(n^γ)$ for $0\leqγ<1/12$.
Random Hermitian matrices with a source term arise, for instance, in the study of non-intersecting Brownian walkers \cite{Adler:2009a, Daems:2007} and sample covariance matrices \cite{Baik:2005}. We consider the case when the $n\times n$ external source matrix has two distinct real eigenvalues: $a$ with multiplicity $r$ and zero with multiplicity $n-r$. The source is small in the sense that $r$ is finite or $r=\mathcal O(n^γ)$, for $0< γ<1$. For a Gaussian potential, Péché \cite{Peche:2006} showed that for $|a|$ sufficiently small (the subcritical regime) the external source has no leading-order effect on the eigenvalues, while for $|a|$ sufficiently large (the supercritical regime) $r$ eigenvalues exit the bulk of the spectrum and behave as the eigenvalues of $r\times r$ Gaussian unitary ensemble (GUE). We establish the universality of these results for a general class of analytic potentials in the supercritical and subcritical regimes.
Recently, a two-matrix-model with a new type of interaction [1] has been introduced and analyzed using bi-orthogonal polynomial techniques. Here we present the complete 1/N^2 expansion for the formal version of this model, following the spirit of [2,3], i.e. the full expansion for the non mixed resolvent correlators and for the free energies.
We reconstruct a rational Lax matrix of size R+1 from its spectral curve (the desingularization of the characteristic polynomial) and some additional data. Using a twisted Cauchy--like kernel (a bi-differential of bi-weight (1-nu,nu)) we provide a residue-formula for the entries of the Lax matrix in terms of bases of dual differentials of weights nu and 1-nu respectively. All objects are described in the most explicit terms using Theta functions. Via a sequence of ``elementary twists'', we construct sequences of Lax matrices sharing the same spectral curve and polar structure and related by conjugations by rational matrices. Particular choices of elementary twists lead to construction of sequences of Lax matrices related to finite--band recurrence relations (i.e. difference operators) sharing the same shape. Recurrences of this kind are satisfied by several types of orthogonal and biorthogonal polynomials. The relevance of formulae obtained to the study of the large degree asymptotics for these polynomials is indicated.
The nonlinear steepest descent method for rank-two systems relies on the notion of g-function. The applicability of the method ranges from orthogonal polynomials (and generalizations) to Painleve transcendents, and integrable wave equations (KdV, NonLinear Schroedinger, etc.). For the case of asymptotics of generalized orthogonal polynomials with respect to varying complex weights we can recast the requirements for the Cauchy-transform of the equilibrium measure into a problem of algebraic geometry and harmonic analysis and completely solve the existence and uniqueness issue without relying on the minimization of a functional. This addresses and solves also the issue of the ``free boundary problem'', determining implicitly the curves where the zeroes of the orthogonal polynomials accumulate in the limit of large degrees and the support of the measure. The relevance to the quasi--linear Stokes phenomenon for Painleve equations is indicated. A numerical algorithm to find these curves in some cases is also explained. Technical note: the animations included in the file can be viewed using Acrobat Reader 7 or higher. Mac users should also install a QuickTime plugin called Flip4Mac. Linux users can extract the embedded animations and play them with an external program like VLC or MPlayer. All trademarks are owned by the respective companies.
We study the massless minimally coupled scalar field on a two--dimensional de Sitter space-time in the setting of axiomatic quantum field theory. We construct the invariant Wightman distribution obtained as the renormalized zero--mass limit of the massive one. Insisting on gauge invariance of the model we construct a vacuum state and a Hilbert space of physical states which are invariant under the action of the whole de Sitter group. We also present the integral expression of the conserved charge which generates the gauge invariance and propose a definition of dual field.
We analyze isomonodromic deformations of rational connections on the Riemann sphere with Fuchsian and irregular singularities. The Fuchsian singularities are allowed to be of arbitrary resonant index; the irregular singularities are also allowed to be resonant in the sense that the leading coefficient matrix at each singularity may have arbitrary Jordan canonical form, with a genericity condition on the Lidskii submatrix of the subleading term. We also give the relevant notion of isomonodromic tau function extending the one of non-resonant deformations introduced by Miwa-Jimbo-Ueno. The tau function is expressed purely in terms of spectral invariants of the matrix of the connection.
We introduce the notion of bilinear moment functional and study their general properties. The analogue of Favard's theorem for moment functionals is proven. The notion of semi-classical bilinear functionals is introduced as a generalization of the corresponding notion for moment functionals and motivated by the applications to multi-matrix random models. Integral representations of such functionals are derived and shown to be linearly independent.