Fredholm Determinant Research Articles

Quasi-Rational and Rational Solutions to the Defocusing Nonlinear Schrödinger Equation

Quasi-rational solutions to the defocusing nonlinear Schrödinger equation (dNLS) in terms of wronskians and Fredholm determinants of order $2N$ depending on $2N-2$ real parameters are given. We get families of quasi-rational solutions to the dNLS equation expressed as a quotient of two polynomials of degree $N(N+1)$ in the variables $x$ and $t$. We present also rational solutions as a quotient of determinants involving certain particular polynomials.

TASEP and generalizations: method for exact solution

The explicit biorthogonalization method, developed in [arXiv:1701.00018] for continuous time TASEP, is generalized to a broad class of determinantal measures which describe the evolution of several interacting particle systems in the KPZ universality class. The method is applied to sequential and parallel update versions of each of the four variants of discrete time TASEP (with Bernoulli and geometric jumps, and with block and push dynamics) which have determinantal transition probabilities; to continuous time PushASEP; and to a version of TASEP with generalized update. In all cases, multipoint distribution functions are expressed in terms of a Fredholm determinant with an explicit kernel involving hitting times of certain random walks to a curve defined by the initial data of the system. The method is further applied to systems of interacting caterpillars, an extension of the discrete time TASEP models which generalizes sequential and parallel updates.

Edge Distribution of Thinned Real Eigenvalues in the Real Ginibre Ensemble

This paper is concerned with the explicit computation of the limiting distribution function of the largest real eigenvalue in the real Ginibre ensemble when each real eigenvalue has been removed independently with constant likelihood. We show that the recently discovered integrable structures in [2] generalize from the real Ginibre ensemble to its thinned equivalent. Concretely, we express the aforementioned limiting distribution function as a convex combination of two simple Fredholm determinants and connect the same function to the inverse scattering theory of the Zakharov–Shabat system. As corollaries, we provide a Zakharov–Shabat evaluation of the ensemble’s real eigenvalue generating function and obtain precise control over the limiting distribution function’s tails. The latter part includes the explicit computation of the usually difficult constant factors.

Zeta and Fredholm determinants of self-adjoint operators

Let L be a self-adjoint invertible operator in a Hilbert space such that L−1 is p-summable. Under a certain discrete dimension spectrum assumption on L, we study the relation between the (regularized) Fredholm determinant, detp(I+z⋅L−1), on the one hand and the zeta regularized determinant, detζ(L+z), on the other. One of the main results is the formuladetζ(L+z)detζ(L)=exp⁡(∑j=1p−1zjj!⋅djdzjlog⁡detζ(L+z)|z=0)⋅detp(I+z⋅L−1). We show that the derivatives djdzjlog⁡detζ(L+z)|z=0 can be expressed in terms of (regularized) zeta values and heat trace coefficients of L. Furthermore, we give a general criterion in terms of the heat trace coefficients (and which is, e.g., fulfilled for large classes of elliptic operators) which guarantees that the constant term in the asymptotic expansion of the Fredholm determinant, log⁡detp(I+z⋅L−1), equals the zeta determinant of L.

Dyson gas on a curved contour

We introduce and study a model of a logarithmic gas with inverse temperature β on an arbitrary smooth closed contour in the plane. This model generalizes Dyson’s gas (the β-ensemble) on the unit circle. We compute the non-vanishing terms of the large N expansion of the free energy (N is the number of particles) by iterating the ‘loop equation’ that is the Ward identity with respect to reparametrizations and dilatation of the contour. We show that the main contribution to the free energy is expressed through the spectral determinant of the Neumann jump operator associated with the contour, or equivalently through the Fredholm determinant of the Neumann–Poincare (double layer) operator. This result connects the statistical mechanics of the Dyson gas to the spectral geometry of the interior and exterior domains of the supporting contour.

On the evaluation of the eigendecomposition of the Airy integral operator

The distributions of the $k$-th largest level at the soft edge scaling limit of Gaussian ensembles are some of the most important distributions in random matrix theory, and their numerical evaluation is a subject of great practical importance. One numerical method for evaluating the distributions uses the fact that they can be represented as Fredholm determinants involving the so-called Airy integral operator. When the spectrum of the integral operator is computed by discretizing it directly, the eigenvalues are known to at most absolute precision. Remarkably, the Airy integral operator is an example of a so-called bispectral operator, which admits a commuting differential operator that shares the same eigenfunctions. In this manuscript, we develop an efficient numerical algorithm for evaluating the eigendecomposition of the Airy integral operator to full relative precision, using the eigendecomposition of the commuting differential operator. This allows us to rapidly evaluate the distributions of the $k$-th largest level to full relative precision rapidly everywhere except in the left tail, where they are computed to absolute precision. In addition, we characterize the eigenfunctions of the Airy integral operator, and describe their extremal properties in relation to an uncertainty principle involving the Airy transform. We observe that the Airy integral operator is fairly universal, and we describe a separate application to Airy beams in optics. Using the eigenfunctions, we compute a finite-energy Airy beam that is optimal, in the sense that the beam is both maximally concentrated, and maximally non-diffracting and self-accelerating.

Large-time and long-distance asymptotics of the thermal correlators of the impenetrable anyonic lattice gas

We study thermal correlation functions of the one-dimensional impenetrable lattice anyons. These correlation functions can be presented as a difference of two Fredholm determinants. To describe the large-time and long-distance behavior of these objects, we use the effective form-factor approach. The asymptotic behavior is different in the spacelike and timelike regions. In particular, in the timelike region we observe the additional power factor on top of the exponential decay. We argue that this result is universal as it is related to the discontinuous behavior of the phase shift function of the effective fermions. At particular values of the anyonic parameter, we recover the asymptotics of the spin-spin correlation functions in the $XX$ quantum chain.

The Hilbert L-matrix

We analyze spectral properties of the Hilbert L-matrix(1max⁡(m,n)+ν)m,n=0∞ regarded as an operator Lν acting on ℓ2(N0), for ν∈R, ν≠0,−1,−2,…. The approach is based on a spectral analysis of the inverse of Lν, which is an unbounded Jacobi operator whose spectral properties are deducible in terms of the unit argument F23-hypergeometric functions. In particular, we give answers to two open problems concerning the operator norm of Lν published by L. Bouthat and J. Mashreghi in [Oper. Matrices 15, No. 1 (2021), 47–58]. In addition, several general aspects concerning the definition of an L-operator, its positivity, and Fredholm determinants are also discussed.

Multi-point distribution of discrete time periodic TASEP

We study the one-dimensional discrete time totally asymmetric simple exclusion process with parallel update rules on a spatially periodic domain. A multi-point space-time joint distribution formula is obtained for general initial conditions. The formula involves contour integrals of Fredholm determinants with kernels acting on certain discrete spaces. For a class of initial conditions satisfying certain technical assumptions, we are able to derive large-time, large-period limit of the joint distribution, under the relaxation time scale t=O(L^{3/2}) when the height fluctuations are critically affected by the finite geometry. The assumptions are verified for the step and flat initial conditions. As a corollary we obtain the multi-point distribution of discrete time TASEP on the whole integer lattice {mathbb {Z}} by taking the period L large enough so that the finite-time distribution is not affected by the boundary. The large time limit for the multi-time distribution of discrete time TASEP on {mathbb {Z}} is then obtained for the step initial condition.

Out-of-equilibrium dynamics of the XY spin chain from form factor expansion

We consider the XY spin chain with arbitrary time-dependent magnetic field and anisotropy. We argue that a certain subclass of Gaussian states, called Coherent Ensemble (CE) following [1], provides a natural and unified framework for out-of-equilibrium physics in this model. We show that all correlation functions in the CE can be computed using form factor expansion and expressed in terms of Fredholm determinants. In particular, we present exact out-of-equilibrium expressions in the thermodynamic limit for the previously unknown order parameter 1-point function, dynamical 2-point function and equal-time 3-point function.

KP governs random growth off a 1-dimensional substrate

Abstract 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.

Exact WKB methods in SU(2) Nf = 1

We study in detail the Schrödinger equation corresponding to the four dimensional SU(2) mathcal{N} = 2 SQCD theory with one flavour. We calculate the Voros symbols, or quantum periods, in four different ways: Borel summation of the WKB series, direct computation of Wronskians of exponentially decaying solutions, the TBA equations of Gaiotto-Moore-Neitzke/Gaiotto, and instanton counting. We make computations by all of these methods, finding good agreement. We also study the exact quantization condition for the spectrum, and we compute the Fredholm determinant of the inverse of the Schrödinger operator using the TS/ST correspondence and Zamolodchikov’s TBA, again finding good agreement. In addition, we explore two aspects of the relationship between singularities of the Borel transformed WKB series and BPS states: BPS states of the 4d theory are related to singularities in the Borel transformed WKB series for the quantum periods, and BPS states of a coupled 2d+4d system are related to singularities in the Borel transformed WKB series for local solutions of the Schrödinger equation.

Multi-Parametric Families of Real and Non Singular Solutions of the Kadomtsev-Petviasvili I Equation

Multi-parametric solutions to the Kadomtsev-Petviashvili equation (KPI) in terms of Fredholm determinants are constructed in function of exponentials. A representation of these solutions as a quotient of wronskians of order $2N$ in terms of trigonometric functions is deduced. All these solutions depend on $2N-1$ real parameters. A third representation in terms of a quotient of two real polynomials depending on $2N-2$ real parameters is given; the numerator is a polynomial of degree $2N(N+1)-2$ in $x$, $y$ and $t$ and the denominator is a polynomial of degree $2N(N+1)$ in $x$, $y$ and $t$. The maximum absolute value is equal to $2(2N+1)^{2}-2$. We explicitly construct the expressions for the first third orders and we study the patterns of their absolute value in the plane $(x,y)$ and their evolution according to time and parameters.\\ It is relevant to emphasize that all these families of solutions are real and non singular.

Application of symmetric analytic functions to spectra of linear operators

The paper is devoted to extension of the theory of symmetric analytic functions on Banach sequence spaces to the spaces of nuclear and $p$-nuclear operators on the Hilbert space. We introduced algebras of symmetric polynomials and analytic functions on spaces of $p$-nuclear operators, described algebraic bases of such algebras and found some connection with the Fredholm determinant of a nuclear operator. In addition, we considered cases of compact and bounded normal operators on the Hilbert space and discussed structures of symmetric polynomials on corresponding spaces.

Quantum Gravity Microstates from Fredholm Determinants.

Various two dimensional quantum gravity theories of Jackiw-Teitelboim (JT) form have descriptions as random matrix models. Such models, treated nonperturbatively, can give an explicit and tractable description of the underlying "microstate" degrees of freedom, which play a prominent role in regimes where the smooth geometrical picture of the physics is inadequate. This is shown using a natural tool, a Fredholm determinant det(1-K), which can be defined explicitly for a wide variety of JT gravity theories. To illustrate the methods, the statistics of the first several energy levels of a nonperturbative definition of JT gravity are constructed explicitly using numerical methods, and the full quenched free energy F_{Q}(T) of the system is computed for the first time. These results are also of relevance to quantum properties of black holes in higher dimensions.

Hecke Triangle Groups, Transfer Operators and Hausdorff Dimension

We consider the family of Hecke triangle groups Gamma _{w} = langle S, T_wrangle generated by the Möbius transformations S : zmapsto -1/z and T_{w} : z mapsto z+w with w > 2. In this case, the corresponding hyperbolic quotient Gamma _{w}backslash {mathbb {H}}^2 is an infinite-area orbifold. Moreover, the limit set of Gamma _w is a Cantor-like fractal whose Hausdorff dimension we denote by delta (w). The first result of this paper asserts that the twisted Selberg zeta function Z_{Gamma _{ w}}(s, rho ) , where rho : Gamma _{w} rightarrow mathrm {U}(V) is an arbitrary finite-dimensional unitary representation, can be realized as the Fredholm determinant of a Mayer-type transfer operator. This result has a number of applications. We study the distribution of the zeros in the half-plane mathrm {Re}(s) > frac{1}{2} of the Selberg zeta function of a special family of subgroups ( Gamma _w^N )_{Nin {mathbb {N}}} of Gamma _w. These zeros correspond to the eigenvalues of the Laplacian on the associated hyperbolic surfaces X_w^N = Gamma _w^N backslash {mathbb {H}}^2. We show that the classical Selberg zeta function Z_{Gamma _w}(s) can be approximated by determinants of finite matrices whose entries are explicitly given in terms of the Riemann zeta function. Moreover, we prove an asymptotic expansion for the Hausdorff dimension delta (w) as wrightarrow infty .

The Energy of the Ground State of the Two-Dimensional Hamiltonian of a Parabolic Quantum Well in the Presence of an Attractive Gaussian Impurity

In this article, we provide an expansion (up to the fourth order of the coupling constant) of the energy of the ground state of the Hamiltonian of a quantum mechanical particle moving inside a parabolic well in the x-direction and constrained by the presence of a two-dimensional impurity, modelled by an attractive two-dimensional isotropic Gaussian potential. By investigating the associated Birman–Schwinger operator and exploiting the fact that such an integral operator is Hilbert–Schmidt, we use the modified Fredholm determinant in order to compute the energy of the ground state created by the impurity.

Fredholm determinant representation of the homogeneous Painlevé II τ-function

We formulate the generic τ-function of the homogeneous Painlevé II equation as a Fredholm determinant of an integrable (Its–Izergin–Korepin–Slavnov) operator. The τ-function depends on the isomonodromic time t and two Stokes parameters. The vanishing locus of the τ-function, called the Malgrange divisor is then determined by the zeros of the Fredholm determinant.

The Laplace transform of the integrated Volterra Wishart process

AbstractWe establish an explicit expression for the conditional Laplace transform of the integrated Volterra Wishart process in terms of a certain resolvent of the covariance function. The core ingredient is the derivation of the conditional Laplace transform of general Gaussian processes in terms of Fredholm's determinant and resolvent. Furthermore, we link the characteristic exponents to a system of non‐standard infinite dimensional matrix Riccati equations. This leads to a second representation of the Laplace transform for a special case of convolution kernel. In practice, we show that both representations can be approximated by either closed form solutions of conventional Wishart distributions or finite dimensional matrix Riccati equations stemming from conventional linear‐quadratic models. This allows fast pricing in a variety of highly flexible models, ranging from bond pricing in quadratic short rate models with rich autocorrelation structures, long range dependence and possible default risk, to pricing basket options with covariance risk in multivariate rough volatility models.

Inverse Scattering of the Zakharov-Shabat System Solves the Weak Noise Theory of the Kardar-Parisi-Zhang Equation.

We solve the large deviations of the Kardar-Parisi-Zhang (KPZ) equation in one dimension at short time by introducing an approach which combines field theoretical, probabilistic, and integrable techniques. We expand the program of the weak noise theory, which maps the large deviations onto a nonlinear hydrodynamic problem, and unveil its complete solvability through a connection to the integrability of the Zakharov-Shabat system. Exact solutions, depending on the initial condition of the KPZ equation, are obtained using the inverse scattering method and a Fredholm determinant framework recently developed. These results, explicit in the case of the droplet geometry, open the path to obtain the complete large deviations for general initial conditions.