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Momenta spacing distributions in anharmonic oscillators and the higher order finite temperature Airy kernel
... Much of the recent relevance of Painlevé equations is due to its appearance in random matrix theory, see [50] for an overview of several of these connections. There has been a growing recent interest in integro-differential Painlevé-type equations [24,26,28,32,58,64], and our results place the integro-differential PII as a central universal object in random matrix theory as well. ...
... Recently, Cafasso et al. [28] also obtained an independent proof of the representation (2.10), extending it to more general multiplicative statistics of the Airy 2 point process. Other proofs and extensions of this integro-differential equation have also been recently found in related contexts [24,26,58]. Also, by exploring (2.12) the tail behavior of the KPZ equation has become rigorously accessible in various asymptotic regimes [27,28,32,37]. ...
... It is relatively simple to write a RHP that should be satisfied by this 0 , and we expect it to be related to the KdV hierarchy [33] but with nonstandard initial data. It would be interesting to see if the particular solutions obtained this way reduce to integro-differential hierarchies of Painlevé equations, in the same spirit of the recent works [26,58]. ...
We study multiplicative statistics for the eigenvalues of unitarily-invariant Hermitian random matrix models. We consider one-cut regular polynomial potentials and a large class of multiplicative statistics. We show that in the large matrix limit several associated quantities converge to limits which are universal in both the polynomial potential and the family of multiplicative statistics considered. In turn, such universal limits are described by the integro-differential Painlevé II equation, and in particular they connect the random matrix models considered with the narrow wedge solution to the KPZ equation at any finite time.
... La première est composée des chapitre de 1 à 4, qui visent les objectifs suivants : introduire les objets à la base de cette étude et la motiver, collectionner les résultats principaux qui relient ces objets entre eux et rappeler les méthodes classiques utilisées en littérature pour démontrer ces résultats. La deuxième partie contient les contributions originales prouvées dans les articles [Tar21,BCT21,Ber21], qui se trouvent respectivement du chapitre 5 au 7. ...
... The second part contains instead the original contributions obtained in the works [Tar21,BCT21,BT21], that are distributed in the last three chapters. In particular the thesis is organised as follows : ...
... (4) In Chapter 6 we go through the proof of the main result of [BCT21] : this time we obtain a generalization of the Tracy-Widom formula for a finite temperature version of the higher order Airy kernels together with a particular solutions of an integro-differential Painlevé II hierarchy. Even though the results of this chapter and the previous one are comparable, the proof of the second one requires more complicated techniques. ...
The Painlevé II hierarchy is a sequence of nonlinear ODEs, with the Painlevé II equation as first member. Each member of the hierarchy admits a Lax pair in terms of isomonodromic deformations of a rank 2 system of linear ODEs, with polynomial coefficient for the homogeneous case. It was recently proved that the Tracy-Widom formula for the Hastings-McLeod solution of the homogeneous PII equation can be extended to analogue solutions of the homogeneous PII hierar-chy using Fredholm determinants of operators acting through higher order Airy kernels. These integral operators are used in the theory of determinantal point processes with applications in statistical mechanics and random matrix theory. From this starting point, this PhD thesis explored the following directions. We found a formula of Tracy-Widom type connecting the Fredholm determinants of operators acting through matrix-valued analogues of the higher order Airy kernels withparticular solution of a matrix-valued PII hierarchy. The result is achieved by using a matrix-valued Riemann-Hilbert problem to study these Fredholm determinants and by deriving a block-matrix Lax pair for the relevant hierarchy. We also found another generalization of the Tracy-Widom formula, this time relating the Fredholm determinants of finite-temperature versions of higher order Airy kernels operators to particular solutions of an integro-differential PII hierarchy. In this setting, a suitable operator-valued Riemann-Hilbert problem is used to study the relevant Fredholm determinant. The study of its solution produces in the end an operator-valued Lax pair that naturally encodes an integro-differential Painlevé II hierarchy. From a more geometrical point of view, we analyzed the Poisson-symplectic structure of the monodromy manifolds associated to a system of linear ODEs with polynomial coefficient, also known as Stokes manifolds. For the rank 2 case, we found explicit log canonical coordinates for the symplectic 2-form, forming a cluster algebra of specific type. Moreover, the log-canonical coordinates constructed in this way provide a linearization of the Poisson structure on the Stokes manifolds, first introduced by Flaschka and Newell in their pioneering work of 1981
... Much of the recent relevance of Painlevé equations is due to its appearance in random matrix theory, see [40] for an overview of several of these connections. There has been a growing recent interest in integro-differential Painlevé-type equations [22,23,25,27,46,51], and our results place the integro-differential PII as a central universal object in random matrix theory as well. ...
... Recently, Cafasso, Claeys and Ruzza [25] also obtained an independent proof of the representation (2.10), extending it to more general multiplicative statistics of the Airy 2 point process. Other proofs and extensions of this integro-differential equation have also been recently found in related contexts [22,23,46]. Also, by exploring (2.12) the tail behavior of the KPZ equation has become rigorously accessible in various asymptotic regimes [24,25,27,30]. ...
... It is relatively simple to write a RHP that should be satisfied by this Φ 0 , and we expect it to be related to the KdV hierarchy [28] but with nonstandard initial data. It would be interesting to see if the particular solutions obtained this way reduce to integro-differential hierarchies of Painlevé equations, in the same spirit of the recent works [23,46]. ...
We study multiplicative statistics for the eigenvalues of unitarily-invariant Hermitian random matrix models. We consider one-cut regular polynomial potentials and a large class of multiplicative statistics. We show that in the large matrix limit several associated quantities converge to limits which are universal in both the potential and the family of multiplicative statistics considered. In turn, such universal limits are described by the integro-differential Painlev\'e II equation, and in particular they connect the random matrix models considered with the narrow wedge solution to the KPZ equation at any finite time.
... We will call the collection of equations in (1.6) the vector valued PII hierarchy. Their formulation, from an algebraic point of view, is completely analogous to the one we previously introduced, in collaboration with Thomas Bothner [13], for the integro-differential Painlevé II hierarchy, see also [10]. Note, however, that the results contained in this article cannot be deduced from the ones in [13], because of the assumption of smoothness for the weight function w, see Section 1.3 in loc.cit. ...
... Their formulation, from an algebraic point of view, is completely analogous to the one we previously introduced, in collaboration with Thomas Bothner [13], for the integro-differential Painlevé II hierarchy, see also [10]. Note, however, that the results contained in this article cannot be deduced from the ones in [13], because of the assumption of smoothness for the weight function w, see Section 1.3 in loc.cit. Remark 1.4. ...
... It is easy to show (see for instance [13]) that, for any real t, ...
We prove a Tracy-Widom type formula for the generating function of occupancy numbers on several disjoint intervals of the higher order Airy point processes. The formula is related to a new vector-valued Painlev\'e II hierarchy we define, together with its Lax pair.
... The asymptotic behavior of the p-Airy function is obtained in [36], and the following results, up to some unkown numerical factor a p andã p , holds ...
... Similarly, all the results in Section 4.2 follow immediately by replacing t 1 with T . Moreover, straightforward generalization of the results for Jordan quiver to the generalized clover quiver leads to 36) and the parameter λ p is obtained as ...
The generating functions for the supersymmetric indices of the gauge theory such as superconformal index are often represented in terms of the unitary matrix integrals with double trace potential. In the limit of weak interactions between the eigenvalues, they can be approximated by the matrix models with the single-trace potential, i.e. the generalized Gross-Witten-Wadia model. In this work, the perturbative and non-perturbative aspects of the generic multi-critical unitary matrix models are studied by adopting the integrable operator formalism, and the multi-critical generalization of the Tracy-Widom distribution in the context of random partitions. We obtain the universal results for the multi-critical model in the weak and strong coupling phases. The free energy of the instanton sector in the weak coupling regime, and the genus expansion of the free energy in the strong coupling regime are explicitly computed and the universal multi-critical phase structure of the model is explored. Finally, we apply our results in concrete examples of supersymmetric indices of gauge theories.
... x log det(Id L 2 (D∪D) − K). [14] and A. Krajenbrink [39] enlarged, in a different direction with respect to our case, the class of Hankel composition operators to obtain new class of solutions of the modified Korteweg di Vries equation. Applications are obtained in [15], [17]. 4 Step-like oscillatory initial data For certain class of domains D called generalized quadrature domains and β analytic, the ∂problem can be reduced to a Riemann problem. ...
We consider soliton gas solutions of the Focusing Nonlinear Schr\"odinger (NLS) equation, where the point spectrum of the Zakharov-Shabat linear operator condensate in a bounded domain in the upper half-plane. We show that the corresponding inverse scattering problem can be formulated as a -problem on the domain. We prove the existence of the solution of this -problem by showing that the -function of the problem (a Fredholm determinant) does not vanish. We then represent the solution of the NLS equation via the of the - problem. Finally we show that, when the domain is an ellipse and the density of solitons is analytic, the initial datum of the Cauchy problem is asymptotically step-like oscillatory, and it is described by a periodic elliptic function as while it vanishes exponentially fast as .
... We choose to work with the algebraic Tracy-Widom method in the derivation of theorem 6.7 for ease of presentation. The same approach was used in [1] in the derivation of the aforementioned integro-differential Painlevé-II connection and only later on it was shown in [6,8] that operator-valued Riemann-Hilbert techniques yield the same result. We expect that the operator-valued Riemann-Hilbert methodology can also be applied to (5.10), (6.1) and it would yield (1.38), (1.39) through the compatibility of a suitable operator-valued Lax pair. ...
We show that the distribution of bulk spacings between pairs of adjacent eigenvalue real parts of a random matrix drawn from the complex elliptic Ginibre ensemble is asymptotically given by a generalization of the Gaudin-Mehta distribution, in the limit of weak non-Hermiticity. The same generalization is expressed in terms of an integro-differential Painlevé function and it is shown that the generalized Gaudin-Mehta distribution describes the crossover, with increasing degree of non-Hermiticity, from Gaudin-Mehta nearest-neighbor bulk statistics in the Gaussian Unitary Ensemble to Poisson gap statistics for eigenvalue real parts in the bulk of the Complex Ginibre Ensemble.
... First, it was proved in a series of recent papers [1,7,8,14] in slightly different forms, that with v σ (s) := ∂ 2 s log j σ (s), (1.14) one has v σ (s) = − R ϕ σ (λ; s) 2 σ (λ)dλ, σ (λ) := dσ (λ) dλ , (1.15) where ϕ σ (λ; s) solves the Stark equation 1 ∂ 2 s + 2v σ (s) − s ϕ(λ; s) = λ ϕ(λ; s). More precisely, ϕ σ (λ; s) is the unique solution to the Stark boundary value problem (1.19) in terms of the function ϕ σ (λ; X, T ) ...
We study Jánossy densities of a randomly thinned Airy kernel determinantal point process. We prove that they can be expressed in terms of solutions to the Stark and cylindrical Korteweg–de Vries equations; these solutions are Darboux tranformations of the simpler ones related to the gap probability of the same thinned Airy point process. Moreover, we prove that the associated wave functions satisfy a variation of Amir–Corwin–Quastel’s integro-differential Painlevé II equation. Finally, we derive tail asymptotics for the relevant solutions to the cylindrical Korteweg–de Vries equation and show that they decompose asymptotically into a superposition of simpler solutions.
... For a suitable class of functions σ , it was shown first in [1] (see also [6,7,9,23]) that (1.16) where ϕ σ solves the integro-differential Painlevé II equation ...
We study a family of Fredholm determinants associated to deformations of the sine kernel, parametrized by a weight function w. For a specific choice of w, this kernel describes bulk statistics of finite temperature free fermions. We establish a connection between these determinants and a system of integro-differential equations generalizing the fifth Painlevé equation, and we show that they allow us to solve an integrable PDE explicitly for a large class of initial data.
... Recently Bothner [6] and A. Krajenbrink [26] enlarged the class of Hankel composition operators that can be studied via Riemann-Hilbert problems. Applications are obtained in [7], [8]. ...
We develop the theory of integrable operators acting on a domain of the complex plane with smooth boundary in analogy with the theory of integrable operators acting on contours of the complex plane. We show how the resolvent operator is obtained from the solution of a -problem in the complex plane. When such a -problem depends on auxiliary parameters we define its Malgrange one form in analogy with the theory of isomonodromic problems. We show that the Malgrange one form is closed and coincides with the exterior logarithmic differential of the Hilbert-Carleman determinant of the operator . With suitable choices of the setup we show that the Hilbert-Carleman determinant is a -function of the Kadomtsev-Petviashvili (KP) or nonlinear Schr\"odinger hierarchies.
... By integrating (1.10) twice, one further obtains the following Tracy-Widom type formula This result has been extended to several disjoint intervals in [12], which is related to a vectorvalued Painlevé II hierarchy. We also refer to [7] for the studies of higher order Airy process at finite temperature, where the law is governed by a Painlevé II integro-differential hierarchy. ...
We study the one-parameter family of Fredholm determinants det ( I − ρ 2 K n , x ) , ρ ∈ R , where K n , x stands for the integral operator acting on L 2 ( x , + ∞ ) with the higher order Airy kernel. This family of determinants represents a new universal class of distributions which is a higher order analogue of the classical Tracy–Widom distribution. Each of the determinants admits an integral representation in terms of a special real solution to the n th member of the Painlevé II hierarchy. Using the Riemann–Hilbert approach, we establish asymptotics of the determinants and the associated higher order Painlevé II transcendents as x → − ∞ for 0 < | ρ | < 1 and | ρ | > 1 , respectively. In the case of 0 < | ρ | < 1 , we are able to calculate the constant term in the asymptotic expansion of the determinants, while for | ρ | > 1 , the relevant asymptotics exhibit singular behaviours. Applications of our results are also discussed, which particularly include asymptotic statistical properties of the counting function for the random point process defined by the higher order Airy kernel.
... By integrating (1.10) twice, one further obtains the following Tracy-Widom type formula This result has been extended to several disjoint intervals in [12], which is related to a vectorvalued Painlevé II hierarchy. We also refer to [7] for the studies of higher order Airy process at finite temperature, where the law is governed by a Painlevé II integro-differential hierarchy. ...
We study the one-parameter family of Fredholm determinants , , where stands for the integral operator acting on with the higher order Airy kernel. This family of determinants represents a new universal class of distributions which is a higher order analogue of the classical Tracy-Widom distribution. Each of the determinants admits an integral representation in terms of a special real solution to the n-th member of the Painlev\'{e} II hierarchy. Using the Riemann-Hilbert approach, we establish asymptotics of the determinants and the associated higher order Painlev\'{e} II transcendents as for and , respectively. In the case of , we are able to calculate the constant term in the asymptotic expansion of the determinants, while for , the relevant asymptotics exhibit singular behaviors. Applications of our results are also discussed, which particularly include asymptotic statistical properties of the counting function for the random point process defined by the higher order Airy kernel.
We develop the theory of integrable operators K acting on a domain of the complex plane with smooth boundary in analogy with the theory of integrable operators acting on contours of the complex plane. We show how the resolvent operator is obtained from the solution of a ∂― -problem in the complex plane. When such a ∂― -problem depends on auxiliary parameters we define its Malgrange one form in analogy with the theory of isomonodromic problems. We show that the Malgrange one form is closed and coincides with the exterior logarithmic differential of the Hilbert–Carleman determinant of the operator K . With suitable choices of the setup we show that the Hilbert–Carleman determinant is a τ-function of the Kadomtsev–Petviashvili (KP) or nonlinear Schrödinger hierarchies.
The focus of this paper is on the distribution function of the rightmost eigenvalue for the complex elliptic Ginibre ensemble in the limit of weak non-Hermiticity. We show how the limiting distribution function can be expressed in terms of an integro-differential Painlevé-II function and how the same captures the nontrivial transition between Poisson and Airy point process extreme value statistics as the degree of non-Hermiticity decreases. Our most explicit new results concern the tail asymptotics of the limiting distribution function. For the right tail we compute the leading order asymptotics uniformly in the degree of non-Hermiticity, for the left tail we compute it close to Hermiticity.
We introduce multicritical Schur measures, which are probability laws on integer partitions which give rise to non-generic fluctuations at their edge. They are in the same universality classes as one-dimensional momentum-space models of free fermions in flat confining potentials, studied by Le Doussal, Majumdar and Schehr. These universality classes involve critical exponents of the form 1/(2m+1)1/(2m+1), with m a positive integer, and asymptotic distributions given by Fredholm determinants constructed from higher order Airy kernels, extending the generic Tracy–Widom GUE distribution recovered for m=1m=1. We also compute limit shapes for the multicritical Schur measures, discuss the finite temperature setting, and exhibit an exact mapping to the multicritical unitary matrix models previously encountered by Periwal and Shevitz.
We study the finite‐temperature deformation of the discrete Bessel point process. We show that its largest particle distribution satisfies a reduction of the 2D Toda equation, as well as a discrete version of the integro‐differential Painlevé II equation of Amir–Corwin–Quastel, and we compute initial conditions for the Poissonization parameter equal to 0. As proved by Betea and Bouttier, in a suitable continuum limit the last particle distribution converges to that of the finite‐temperature Airy point process. We show that the reduction of the 2D Toda equation reduces to the Korteweg–de Vries equation, as well as the discrete integro‐differential Painlevé II equation reduces to its continuous version. Our approach is based on the discrete analogue of Its–Izergin–Korepin–Slavnov theory of integrable operators developed by Borodin and Deift.
We prove a Tracy-Widom type formula for the generating function of occupancy numbers on several disjoint intervals of the higher order Airy point processes. The formula is related to a new vector-valued Painlevé II hierarchy we define, together with its Lax pair.
We study a family of unbounded solutions to the Korteweg–de Vries equation which can be constructed as log-derivatives of deformed Airy kernel Fredholm determinants, and which are connected to an integro-differential version of the second Painlevé equation. The initial data of the Korteweg–de Vries solutions are well-defined for , but not for , where the solutions behave like as , and hence would be well-defined as solutions of the cylindrical Korteweg–de Vries equation. We provide uniform asymptotics in x as ; for they involve an integro-differential analogue of the Painlevé V equation. A special case of our results yields improved estimates for the tails of the narrow wedge solution to the Kardar–Parisi–Zhang equation.
The logarithmic derivative of the marginal distributions of randomly fluctuating interfaces in one dimension on a large scale evolve according to the Kadomtsev–Petviashvili (KP) equation. This is derived algebraically from a Fredholm determinant obtained in [MQR17] for the Kardar–Parisi–Zhang (KPZ) fixed point as the limit of the transition probabilities of TASEP, a special solvable model in the KPZ universality class. The Tracy–Widom distributions appear as special self-similar solutions of the KP and Korteweg–de Vries equations. In addition, it is noted that several known exact solutions of the KPZ equation also solve the KP equation.
This article is firstly a historic review of the theory of Riemann–Hilbert problems with particular emphasis placed on their original appearance in the context of Hilbert’s 21st problem and Plemelj’s work associated with it. The secondary purpose of this note is to invite a new generation of mathematicians to the fascinating world of Riemann–Hilbert techniques and their modern appearances in nonlinear mathematical physics. We set out to achieve this goal with six examples, including a new proof of the integro-differential Painlevé-II formula of Amir et al (2011 Commun. Pure Appl. Math. 64 466–537) that enters in the description of the Kardar–Parisi–Zhang crossover distribution. Parts of this text are based on the author’s Szegő prize lecture at the 15th International Symposium on Orthogonal Polynomials, Special Functions and Applications (OPSFA) in Hagenberg, Austria.
We study a family of unbounded solutions to the Korteweg–de Vries equation which can be constructed as log-derivatives of deformed Airy kernel Fredholm determinants, and which are connected to an integro-differential version of the second Painlevé equation. The initial data of the Korteweg–de Vries solutions are well-defined for , but not for , where the solutions behave like as , and hence would be well-defined as solutions of the cylindrical Korteweg–de Vries equation. We provide uniform asymptotics in x as ; for they involve an integro-differential analogue of the Painlevé V equation. A special case of our results yields improved estimates for the tails of the narrow wedge solution to the Kardar–Parisi–Zhang equation.
We establish the universal edge scaling limit of random partitions with the infinite-parameter distribution called the Schur measure. We explore the asymptotic behavior of the wave function, which is a building block of the corresponding kernel, based on the Schrödinger-type differential equation. We show that the wave function is in general asymptotic to the Airy function and its higher-order analogs in the edge scaling limit. We construct the corresponding higher-order Airy kernel and the Tracy–Widom distribution from the wave function in the scaling limit and discuss its implication to the multicritical phase transition in the large-size matrix model. We also discuss the limit shape of random partitions through the semi-classical analysis of the wave function.
As Fredholm determinants are more and more frequent in the context of stochastic integrability, we unveil the existence of a common framework in many integrable systems where they appear. This consists in a quasi-universal hierarchy of equations, partly unifying an integro-differential generalization of the Painlevé II hierarchy, the finite-time solutions of the Kardar–Parisi–Zhang equation, multi-critical fermions at finite temperature and a notable solution to the Zakharov–Shabat system associated to the largest real eigenvalue in the real Ginibre ensemble. As a byproduct, we obtain the explicit unique solution to the inverse scattering transform of the Zakharov–Shabat system in terms of a Fredholm determinant.
We revisit the periodic Schur process introduced by Borodin in 2007. Our contribution is threefold. First, we provide a new simpler derivation of its correlation functions via the free fermion formalism. In particular, we shall see that the process becomes determinantal by passing to the grand canonical ensemble, which gives a physical explanation to Borodin’s “shift-mixing” trick. Second, we consider the edge scaling limit in the simplest nontrivial case, corresponding to a deformation of the poissonized Plancherel measure on partitions. We show that the edge behavior is described by the universal finite-temperature Airy kernel, which was previously encountered by Johansson and Le Doussal et al. in other models, and whose extreme value statistics interpolates between the Tracy–Widom GUE and the Gumbel distributions. We also define and prove convergence for a stationary extension of our model. Finally, we compute the correlation functions for a variant of the periodic Schur process involving strict partitions, Schur’s P and Q functions, and neutral fermions.
The purpose of this paper is to push forward the theory of operator-valued
Riemann Hilbert problems and demonstrate their effectiveness in respect to the
implementation of a non-linear steepest descent method \textit{\'{a} la}
Deift-Zhou. In the present paper, we demonstrate that the operator-valued
Riemann--Hilbert problem arising in the characterisation of so-called
c-shifted integrable integral operators allows one to extract the large-x
asymptotics of the Fredholm determinant associated with such operators.
In this paper we give an algorithmic method of deriving the Lax pair for the modified Korteweg-de Vries hierarchy. For each n, the compatibility condition gives the n-th member of the hierarchy, rather than its derivative. A direct conse-quence of this is that we obtain the isomonodromy problem for the second Painlevé hierarchy, which is derived through a scaling reduction. Résumé (La paire de Lax de la hiérarchie mKdV). — Dans cet article, nous pré-sentons une méthode algorithmique pour le calcul de la paire de Lax de la hiérarchie de Korteweg-de Vries modifiée. Pour tout n, la condition de com-patibilité fournit le nì eme membre de la hiérarchie lui-même et non pas sa dérivée. Grâcè a une réduction par l'action du groupe de similarité, nous en déduisons unprobì eme d'isomonodromie pour ladeuxì eme hiérarchie de Pain-levé.
The Tracy-Widom distribution that has been much studied in recent years can be thought of as an extreme value distribution.
We discuss interpolation between the classical extreme value distribution exp(−exp(−x)), the Gumbel distribution, and the Tracy-Widom distribution. There is a family of determinantal processes whose edge behaviour
interpolates between a Poisson process with density exp(−x) and the Airy kernel point process. This process can be obtained as a scaling limit of a grand canonical version of a random
matrix model introduced by Moshe, Neuberger and Shapiro. We also consider the deformed GUE ensemble,
+ VM=M_0+\sqrt{2S} V, with M
0 diagonal with independent elements and V from GUE. Here we do not see a transition from Tracy-Widom to Gumbel, but rather a transition from Tracy-Widom to Gaussian.
We survey recent results on determinantal processes, random growth, random tilings and their relation to random matrix theory.
Scaling level-spacing distribution functions in the ``bulk of the spectrum'' in random matrix models of hermitian matrices and then going to the limit , leads to the Fredholm determinant of the sine kernel . Similarly a scaling limit at the ``edge of the spectrum'' leads to the Airy kernel . In this paper we derive analogues for this Airy kernel of the following properties of the sine kernel: the completely integrable system of P.D.E.'s found by Jimbo, Miwa, M{\^o}ri and Sato; the expression, in the case of a single interval, of the Fredholm determinant in terms of a Painlev{\'e} transcendent; the existence of a commuting differential operator; and the fact that this operator can be used in the derivation of asymptotics, for general n, of the probability that an interval contains precisely n eigenvalues. Comment: 35 pages, LaTeX document using REVTEX macros
In previous work the authors considered the asymmetric simple exclusion process on the integer lattice in the case of step initial condition, particles beginning at the positive integers. There it was shown that the probability distribution for the position of an individual particle is given by an integral whose integrand involves a Fredholm determinant. Here we use this formula to obtain three asymptotic results for the positions of these particles. In one an apparently new distribution function arises and in another the distribuion function F_2 arises. The latter extends a result of Johansson on TASEP to ASEP. Comment: 29 pages. Version 2 has a new title and adds asymptotics in a third regime
A special class of four-point correlation functions in the maximally supersymmetric Yang-Mills theory is given by the square of the Fredholm determinant of a generalized Bessel kernel. In this note, we re-express its logarithmic derivatives in terms of a two-dimensional Riemann-Hilbert problem. We solve the latter in the null limit making use of the Deift-Zhou steepest descent. We reproduce the exact octagonal anomalous dimension in 't Hooft coupling and provide its novel formulation as the convolution of a non-linear quasiclassical phase with the Fermi distribution in the limit of the infinite chemical potential.
We consider Fredholm determinants of matrix Hankel operators associated to matrix versions of the n-th Airy functions. Using the theory of integrable operators, we relate them to a fully noncommutative Painlevé II hierarchy, defined through a matrix-valued version of the Lenard operators. In particular, the Riemann-Hilbert techniques used to study these integrable operators allows to find a Lax pair for each member of the hierarchy. Finally, the coefficients of the Lax matrices are explicitly written in terms of the matrix-valued Lenard operators and some solutions of the hierarchy are written in terms of Fredholm determinants of the square of the matrix Airy Hankel operators.
The quantum inverse scattering method is a means of finding exact solutions of two-dimensional models in quantum field theory and statistical physics (such as the sine-Gordon equation or the quantum non-linear Schrödinger equation). These models are the subject of much attention amongst physicists and mathematicians. The present work is an introduction to this important and exciting area. It consists of four parts. The first deals with the Bethe ansatz and calculation of physical quantities. The authors then tackle the theory of the quantum inverse scattering method before applying it in the second half of the book to the calculation of correlation functions. This is one of the most important applications of the method and the authors have made significant contributions to the area. Here they describe some of the most recent and general approaches and include some new results. The book will be essential reading for all mathematical physicists working in field theory and statistical physics.
We study Fredholm determinants of a class of integral operators, whose kernels can be expressed as double contour integrals of a special type. Such Fredholm determinants appear in various random matrix and statistical physics models. We show that the logarithmic derivatives of the Fredholm determinants are directly related to solutions of the Painlevé II hierarchy. This confirms and generalizes a recent conjecture by Le Doussal, Majumdar, and Schehr [20]. In addition, we obtain asymptotics at for the Painlevé transcendents and large gap asymptotics for the corresponding point processes.
We compute the joint statistics of the momenta of N non-interacting fermions in a trap, near the Fermi edge, with a particular focus on the largest one . For a 1d harmonic trap, momenta and positions play a symmetric role and hence, the joint statistics of momenta is identical to that of the positions. In particular, , as , is distributed according to the Tracy-Widom distribution. Here we show that novel "momentum edge statistics" emerge when the curvature of the potential vanishes, i.e. for "flat traps" near their minimum, with and . These are based on generalisations of the Airy kernel that we obtain explicitly. The fluctuations of are governed by new universal distributions determined from the n-th member of the second Painlev\'e hierarchy of non-linear differential equations, with connections to multicritical random matrix models. Finite temperature extensions and possible experimental signatures in cold atoms are discussed.
We study analytically the Wigner function of N noninteracting fermions trapped in a smooth confining potential in d dimensions. At zero temperature, is constant over a finite support in the phase space and vanishes outside. Near the edge of this support, we find a universal scaling behavior of for large N. The associated scaling function is independent of the precise shape of the potential as well as the spatial dimension d. We further generalize our results to finite temperature . We show that there exists a low temperature regime where is an energy scale that depends on N and the confining potential , where the Wigner function at the edge again takes a universal scaling form with a b-dependent scaling function. This temperature dependent scaling function is also independent of the potential as well as the dimension d. Our results generalize to any and the d=1 and T=0 results obtained by Bettelheim and Wiegman [Phys. Rev. B , 085102 (2011)].
The authors initially planned to write an article describing the origins and devel opments of the theory of Fredholm operators and to present their recollections of this topic. We started to read again classical papers and we were sidetracked by the literature concerned with the theory and applications of traces and determi nants of infinite matrices and integral operators. We were especially impressed by the papers of Poincare, von Koch, Fredholm, Hilbert and Carleman, as well as F. Riesz's book on infinite systems of linear equations. Consequently our plans were changed and we decided to write a paper on the history of determinants of infi nite matrices and operators. During the preparation of our paper we realized that many mathematical questions had to be answered in order to gain a more com plete understanding of the subject. So, we changed our plans again and decided to present the subject in a more advanced form which would satisfy our new require ments. This whole process took between four and five years of challenging, but enjoyable work. This entailed the study of the appropriate relatively recent results of Grothendieck, Ruston, Pietsch, Hermann Konig and others. After the papers [GGK1] and [GGK2] were published, we saw that the written material could serve as the basis of a book.
We derive the local statistics of the canonical ensemble of free fermions in a quadratic potential well at finite temperature, as the particle number approaches infinity. This free fermion model is equivalent to a random matrix model proposed by Moshe, Neuberger and Shapiro. Limiting behaviors obtained before for the grand canonical ensemble are observed in the canonical ensemble: We have at the edge the phase transition from the Tracy--Widom distribution to the Gumbel distribution via the Kardar-Parisi-Zhang (KPZ) crossover distribution, and in the bulk the phase transition from the sine point process to the Poisson point process. A similarity between this model and a class of models in the KPZ universality class is explained. We also derive the multi-time correlation functions and the multi-time gap probability formulas for the free fermions along the imaginary time.
We study a system of N non-interacting spin-less fermions trapped in a confining potential, in arbitrary dimensions d and arbitrary temperature T. The presence of the trap introduces an edge where the average density of fermions vanishes. Far from the edge, near the center of the trap (the so called "bulk regime"), physical properties of the fermions have traditionally been understood using the Local Density Approximation. However, this approximation drastically fails near the edge where the density vanishes. In this paper we show that, even near the edge, novel universal properties emerge, independently of the details of the confining potential. We show that for large N, these fermions in a confining trap, in arbitrary dimensions and at finite temperature, form a determinantal point process. As a result, any n-point correlation function can be expressed as an determinant whose entry is called the kernel. Near the edge, we derive the large N scaling form of the kernels. In d=1 and T=0, this reduces to the so called Airy kernel, that appears in the Gaussian Unitary Ensemble (GUE) of random matrix theory. In d=1 and we show a remarkable connection between our kernel and the one appearing in the 1+1-dimensional Kardar-Parisi-Zhang equation at finite time. Consequently our result provides a finite T generalization of the Tracy-Widom distribution, that describes the fluctuations of the rightmost fermion at T=0. In and , while the connection to GUE no longer holds, the process is still determinantal whose analysis provides a new class of kernels, generalizing the 1d Airy kernel at T=0 obtained in random matrix theory. Some of our finite temperature results should be testable in present-day cold atom experiments, most notably our detailed predictions for the temperature dependence of the fluctuations near the edge.
In this paper we consider a probability distribution on plane partitions, which arises as a one-parameter generalization of the q^{volume} measure. This generalization is closely related to the classical multivariate Hall-Littlewood polynomials, and it was first introduced by Vuletic. We prove that as the plane partitions become large (q goes to 1, while the Hall-Littlewood parameter t is fixed), the scaled bottom slice of the random plane partition converges to a deterministic limit shape, and that one-point fluctuations around the limit shape are asymptotically given by the GUE Tracy-Widom distribution. On the other hand, if t simultaneously converges to its own critical value of 1, the fluctuations instead converge to the one-dimensional Kardar-Parisi-Zhang (KPZ) equation with the so-called narrow wedge initial data. The algebraic part of our arguments is closely related to the formalism of Macdonald processes. The analytic part consists of detailed asymptotic analysis of the arising Fredholm determinants.
Brownian motion is a continuum scaling limit for a wide class of random processes, and there has been great success in developing a theory for its properties (such as distribution functions or regularity) and expanding the breadth of its universality class. Over the past 25 years a new universality class has emerged to describe a host of important physical and probabilistic models (including one-dimensional interface growth processes, interacting particle systems and polymers in random environments) which display characteristic, though unusual, scalings and new statistics. This class is called the Kardar–Parisi–Zhang (KPZ) universality class and underlying it is, again, a continuum object — a non-linear stochastic partial differential equation — known as the KPZ equation.
The purpose of this survey is to explain the context for, as well as the content of a number of mathematical breakthroughs which have culminated in the derivation of the exact formula for the distribution function of the KPZ equation started with narrow wedge initial data. In particular we emphasize three topics: (1) The approximation of the KPZ equation through the weakly asymmetric simple exclusion process; (2) The derivation of the exact one-point distribution of the solution to the KPZ equation with narrow wedge initial data; (3) Connections with directed polymers in random media.
As the purpose of this article is to survey and review, we make precise statements but provide only heuristic arguments with indications of the technical complexities necessary to make such arguments mathematically rigorous.
A systematic treatment is presented of the principal mathematical
problems in nonrelativistic quantum mechanics that are associated with
the study of the Schroedinger equation. In particular, attention is
given to the spectral theory of one-dimensional and multidimensional
Schroedinger operators, scattering theory, and the method of functional
integrals. The discussion also covers operator symbols, operators in
Hilbert space, Sobolev spaces, and elliptic equations.
The paper demonstrates that rational solutions can be found to the classical transcendents of Painleve equations and generalizes the solutions of Gambier to many other cases. Explicit solutions of Painleve equations 2 and 4 are constructed with Baecklund transformations, and degenerate cases of equations 3 and 5 are solved independently. Similarity solutions of the Korteweg-de Vries equations are considered; results obtained for the second Painleve equation are extended to higher order equations.
An article published recently by one of the authors (JNE) presents closed form solutions for zero-curvature representations of the vector nonlinear Schrödinger hierarchy. Several of these results are confirmed computationally but left unproven. In this article we begin by providing strict algebraic proofs of these results. The forms of the hierarchy’s associated spectral curves are investigated and proven to have only odd genus without the introduction of integration constants. We admit specific non-zero constants of integration to the hierarchy and show that even genus curves may be introduced in a very straightforward manner, thereby accessing the full family of finite gap solutions to the vector nonlinear Schrödinger equation. Finally an original construction of the infinite set of conserved densities of the hierarchy is given.
In this paper we are concerned with hierarchies of rational solutions and associated polynomials for the second Painlevé equation (PII) and the equations in the PII hierarchy which is derived from the modified Korteweg–de Vries hierarchy. These rational solutions of PII are expressible as the logarithmic derivative of special polynomials, the Yablonskii–Vorob'ev polynomials. The structure of the roots of these Yablonskii–Vorob'ev polynomials is studied and it is shown that these have a highly regular triangular structure. Further, the properties of the Yablonskii–Vorob'ev polynomials are compared and contrasted with those of classical orthogonal polynomials. We derive the special polynomials for the second and third equations of the PII hierarchy and give a representation of the associated rational solutions in the form of determinants through Schur functions. Additionally the analogous special polynomials associated with rational solutions and representation in the form of determinants are conjectured for higher equations in the PII hierarchy. The roots of these special polynomials associated with rational solutions for the equations of the PII hierarchy also have a highly regular structure.
This paper contains an exposition of both recent and rather old results on determinantal random point fields. We begin with some general theorems including proofs of necessary and sufficient conditions for the existence of a determinantal random point field with Hermitian kernel and of a criterion for weak convergence of its distribution. In the second section we proceed with examples of determinantal random fields in quantum mechanics, statistical mechanics, random matrix theory, probability theory, representation theory, and ergodic theory. In connection with the theory of renewal processes, we characterize all Hermitian determinantal random point fields on and with independent identically distributed spacings. In the third section we study translation-invariant determinantal random point fields and prove the mixing property for arbitrary multiplicity and the absolute continuity of the spectra. In the last section we discuss proofs of the central limit theorem for the number of particles in a growing box and of the functional central limit theorem for the empirical distribution function of spacings.
We study Fredholm determinants related to a family of kernels that describe the edge eigenvalue behavior in unitary random matrix models with critical edge points. The kernels are natural higher-order analogues of the Airy kernel and are built out of functions associated with the Painlevé I hierarchy. The Fredholm determinants related to those kernels are higher-order generalizations of the Tracy-Widom distribution. We give an explicit expression for the determinants in terms of a distinguished smooth solution to the Painlevé II hierarchy. In addition, we compute large gap asymptotics for the Fredholm determinants. © 2009 Wiley Periodicals, Inc.
A new and simpler construction of the family of rational solutions of the Korteweg-deVries equation is given. This construction is related to a factorization of the Sturm-Liouville operators into first order operators and a new deformation problem for the latter. In the final section the spectral representation for the corresponding complex potentials is discussed.
We consider two models for directed polymers in space-time independent random media (the O'Connell-Yor semidiscrete directed polymer and the continuum directed random polymer) at positive temperature and prove their KPZ universality via asymptotic analysis of exact Fredholm determinant formulas for the Laplace transform of their partition functions. In particular, we show that for large time τ, the probability distributions for the free energy fluctuations, when rescaled by τ [superscript 1 over 3], converges to the GUE Tracy-Widom distribution.
We also consider the effect of boundary perturbations to the quenched random media on the limiting free energy statistics. For the semidiscrete directed polymer, when the drifts of a finite number of the Brownian motions forming the quenched random media are critically tuned, the statistics are instead governed by the limiting Baik–Ben Arous–Péché distributions from spiked random matrix theory. For the continuum polymer, the boundary perturbations correspond to choosing the initial data for the stochastic heat equation from a particular class, and likewise for its logarithm—the Kardar-Parisi-Zhang equation. The Laplace transform formula we prove can be inverted to give the one-point probability distribution of the solution to these stochastic PDEs for the class of initial data.
We consider the gap probability for the Pearcey and Airy processes; we set up
a Riemann--Hilbert approach (different from the standard one) whereby the
asymptotic analysis for large gap/large time of the Pearcey process is shown to
factorize into two independent Airy processes using the Deift-Zhou steepest
descent analysis. Additionally we relate the theory of Fredholm determinants of
integrable kernels and the theory of isomonodromic tau function. Using the
Riemann-Hilbert problem mentioned above we construct a suitable Lax pair
formalism for the Pearcey gap probability and re-derive the two nonlinear PDEs
recently found and additionally find a third one not reducible to those.
We consider the solution of the stochastic heat equation
with delta function initial condition
whose logarithm, with appropriate normalization, is the free energy of the continuum directed polymer, or the Hopf-Cole solution of the Kardar-Parisi-Zhang equation with narrow wedge initial conditions.
We obtain explicit formulas for the one-dimensional marginal distributions, the crossover distributions, which interpolate between a standard Gaussian distribution (small time) and the GUE Tracy-Widom distribution (large time). The proof is via a rigorous steepest-descent analysis of the Tracy-Widom formula for the asymmetric simple exclusion process with antishock initial data, which is shown to converge to the continuum equations in an appropriate weakly asymmetric limit. The limit also describes the crossover behavior between the symmetric and asymmetric exclusion processes.
A method for solving certain nonlinear ordinary and partial differential equations is developed. The central idea is to study monodromy preserving deformations of linear ordinary differential equations with regular and irregular singular points. The connections with isospectral deformations and with classical and recent work on monodromy preserving deformations are discussed. Specific new results include the reduction of the general initial value problem for the Painlevé equations of the second type and a special case of the third type to a system of linear singular integral equations. Several classes of solutions are discussed, and in particular the general expression for rational solutions for the second Painlevé equation family is shown to be -d/dx ln(?+/?-), where ?+ and ?- are determinants. We also demonstrate that each of these equations is an exactly integrable Hamiltonian system. The basic ideas presented here are applicable to a broad class of ordinary and partial differential equations; additional results will be presented in a sequence of future papers.
We consider unitary random matrix ensembles on the space of Hermitian n × n matrices M, where the confining potential V
s,t
is such that the limiting mean density of eigenvalues (as n→∞ and s,t→ 0) vanishes like a power 5/2 at a (singular) endpoint of its support. The main purpose of this paper is to prove universality of the eigenvalue correlation kernel in a double scaling limit. The limiting kernel is built out of functions associated with a special solution of the P
I
2 equation, which is a fourth order analogue of the Painlevé I equation. In order to prove our result, we use the well-known connection between the eigenvalue correlation kernel and the Riemann-Hilbert (RH) problem for orthogonal polynomials, together with the Deift/Zhou steepest descent method to analyze the RH problem asymptotically. The key step in the asymptotic analysis will be the construction of a parametrix near the singular endpoint, for which we use the model RH problem for the special solution of the P
I
2 equation.
In addition, the RH method allows us to determine the asymptotics (in a double scaling limit) of the recurrence coefficients of the orthogonal polynomials with respect to the varying weights on . The special solution of the P
I
2 equation pops up in the n
−2/7-term of the asymptotics.
Advanced Complex Analysis. A Comprehensive Course in Analysis, Part 2B
- B Simon
Operator Theory. A Comprehensive Course in Analysis, Part 4
- B Simon
Multicritical random partitions
- D Betea
- J Bouttier
- H Walsh
Mathematical Surveys and Monographs 120
- B Simon