Multiple Contour Integrals Research Articles

Transition probability and total crossing events in the multi-species asymmetric exclusion process

We present explicit formulas for total crossing events in the multi-species asymmetric exclusion process (r-ASEP) with underlying symmetry. In the case of the two-species TASEP these can be derived using an explicit expression for the general transition probability on in terms of a multiple contour integral derived from a nested Bethe ansatz approach. For the general r-ASEP we employ a vertex model approach within which the probability of total crossing can be derived from partial symmetrisation of an explicit high rank rainbow partition function. In the case of r-TASEP, the total crossing probability can be show to reduce to a multiple integral over the product of r determinants. For two-TASEP we additionally derive convenient formulas for cumulative total crossing probabilities using Bernoulli-step initial conditions for particles of type 2 and type 1 respectively.

Multitime Distribution in Discrete Polynuclear Growth

AbstractWe study the multitime distribution in a discrete polynuclear growth model or, equivalently, in directed last‐passage percolation with geometric weights. A formula for the joint multitime distribution function is derived in the discrete setting. It takes the form of a multiple contour integral of a block Fredholm determinant. The asymptotic multitime distribution is then computed by taking the appropriate KPZ‐scaling limit of this formula. This distribution is expected to be universal for models in the Kardar‐Parisi‐Zhang universality class. © 2021 The Authors. Communications on Pure and Applied Mathematics published by Wiley Periodicals LLC.

Correlation functions by separation of variables: the XXX spin chain

We explain how to compute correlation functions at zero temperature within the framework of the quantum version of the Separation of Variables (SoV) in the case of a simple model: the XXX Heisenberg chain of spin 1/2 with twisted (quasi-periodic) boundary conditions. We first detail all steps of our method in the case of anti-periodic boundary conditions. The model can be solved in the SoV framework by introducing inhomogeneity parameters. The action of local operators on the eigenstates are then naturally expressed in terms of multiple sums over these inhomogeneity parameters. We explain how to transform these sums over inhomogeneity parameters into multiple contour integrals. Evaluating these multiple integrals by the residues of the poles outside the integration contours, we rewrite this action as a sum involving the roots of the Baxter polynomial plus a contribution of the poles at infinity. We show that the contribution of the poles at infinity vanishes in the thermodynamic limit, and that we recover in this limit for the zero-temperature correlation functions the multiple integral representation that had been previously obtained through the study of the periodic case by Bethe Ansatz or through the study of the infinite volume model by the q-vertex operator approach. We finally show that the method can easily be generalized to the case of a more general non-diagonal twist: the corresponding weights of the different terms for the correlation functions in finite volume are then modified, but we recover in the thermodynamic limit the same multiple integral representation than in the periodic or anti-periodic case, hence proving the independence of the thermodynamic limit of the correlation functions with respect to the particular form of the boundary twist.

Averages of Products and Ratios of Characteristic Polynomials in Polynomial Ensembles

Polynomial ensembles are a sub-class of probability measures within determinantal point processes. Examples include products of independent random matrices, with applications to Lyapunov exponents, and random matrices with an external field, that may serve as schematic models of quantum field theories with temperature. We first analyse expectation values of ratios of an equal number of characteristic polynomials in general polynomial ensembles. Using Schur polynomials, we show that polynomial ensembles constitute Giambelli compatible point processes, leading to a determinant formula for such ratios as in classical ensembles of random matrices. In the second part, we introduce invertible polynomial ensembles given, e.g. by random matrices with an external field. Expectation values of arbitrary ratios of characteristic polynomials are expressed in terms of multiple contour integrals. This generalises previous findings by Fyodorov, Grela, and Strahov. for a single ratio in the context of eigenvector statistics in the complex Ginibre ensemble.

Loop equation and exact soft anomalous dimension in mathcal{N} = 4 super Yang-Mills

BPS Wilson loops in supersymmetric gauge theories have been the subjects of active research since they are often amenable to exact computation. So far most of the studies have focused on loops that do not intersect. In this paper, we derive exact results for intersecting 1/8 BPS Wilson loops in mathcal{N} = 4 supersymmetric Yang-Mills theory, using a combination of supersymmetric localization and the loop equation in 2d gauge theory. The result is given by a novel matrix-model-like representation which couples multiple contour integrals and a Gaussian matrix model. We evaluate the integral at large N , and make contact with the string worldsheet description at strong coupling. As an application of our results, we compute exactly a small-angle limit (and more generally near-BPS limits) of the cross anomalous dimension which governs the UV divergence of intersecting Wilson lines. The same quantity describes the soft anomalous dimension of scattering amplitudes of W -bosons in the Coulomb branch.

On the Moments of the Moments of the Characteristic Polynomials of Random Unitary Matrices

Denoting by $P_N(A,\theta)=\det(I-Ae^{-i\theta})$ the characteristic polynomial on the unit circle in the complex plane of an $N\times N$ random unitary matrix $A$, we calculate the $k$th moment, defined with respect to an average over $A\in U(N)$, of the random variable corresponding to the $2\beta$th moment of $P_N(A,\theta)$ with respect to the uniform measure $\frac{d\theta}{2\pi}$, for all $k, \beta\in\mathbb{N}$ . These moments of moments have played an important role in recent investigations of the extreme value statistics of characteristic polynomials and their connections with log-correlated Gaussian fields. Our approach is based on a new combinatorial representation of the moments using the theory of symmetric functions, and an analysis of a second representation in terms of multiple contour integrals. Our main result is that the moments of moments are polynomials in $N$ of degree $k^2\beta^2-k+1$. This resolves a conjecture of Fyodorov \& Keating~\cite{fyodorov14} concerning the scaling of the moments with $N$ as $N\rightarrow\infty$, for $k,\beta\in\mathbb{N}$. Indeed, it goes further in that we give a method for computing these polynomials explicitly and obtain a general formula for the leading coefficient.

Six-vertex Models and the GUE-corners Process

Abstract We consider a class of probability distributions on the six-vertex model, which originates from the higher spin vertex models of [13]. We define operators, inspired by the Macdonald difference operators, which extract various correlation functions, measuring the probability of observing different arrow configurations. For the class of models we consider, the correlation functions can be expressed in terms of multiple contour integrals, which are suitable for asymptotic analysis. For a particular choice of parameters we analyze the limit of the correlation functions through the steepest descent method. Combining this asymptotic statement with some new results about Gibbs measures on Gelfand–Tsetlin cones and patterns, we show that the asymptotic behavior of our six-vertex model near the boundary is described by the Gaussian Unitary Ensemble-corners process.

Clustering and the three-point function

We develop analytical methods for computing the structure constant for three heavy operators, starting from the recently proposed hexagon approach. Such a structure constant is a semiclassical object, with the scale set by the inverse length of the operators playing the role of the Planck constant. We reformulate the hexagon expansion in terms of multiple contour integrals and recast it as a sum over clusters generated by the residues of the measure of integration. We test the method on two examples. First, we compute the semi-classical limit of the asymptotic three-point function of heavy operators at any coupling and show that it reproduces part of the results at strong coupling. Second, in the case of one non-BPS and two BPS operators at strong coupling we sum up all wrapping corrections associated with the opposite bridge to the non-trivial operator, or the ‘bottom’ mirror channel. We also give an alternative interpretation of the results in terms of a gas of fermions and show that they can be expressed compactly as an operator-valued super-determinant.

Reflection algebra and functional equations

In this work we investigate the possibility of using the reflection algebra as a source of functional equations. More precisely, we obtain functional relations determining the partition function of the six-vertex model with domain-wall boundary conditions and one reflecting end. The model's partition function is expressed as a multiple-contour integral that allows the homogeneous limit to be obtained straightforwardly. Our functional equations are also shown to give rise to a consistent set of partial differential equations satisfied by the partition function.

Scalar Product of Bethe Vectors from Functional Equations

In this work the scalar product of Bethe vectors for the six-vertex model is studied by means of functional equations. The scalar products are shown to obey a system of functional equations originated from the Yang-Baxter algebra and its solution is given as a multiple contour integral.

A study of polymer knots using a simple knot invariant consisting of multiple contour integrals

In this work the thermodynamic properties of short polymer knots (up to 120 segments) defined on a simple cubic lattice are studied with the help of the Wang–Landau Monte Carlo algorithm. The sampling process is performed using pivot transformations starting from a given seed conformation. Both cases of short-range attractive and repulsive interactions acting on the monomers are considered. The properties of the specific energy, heat capacity and gyration radius of the knots 31,41 and 51 are discussed. It is found that the heat capacity exhibits a sharp peak. If the interactions are attractive, similar peaks have been observed also in single open chains and have been related to the transition from a frozen crystallite state to an expanded coil state. Some other peculiarities of the behavior of the analyzed observables are presented, such as the increase or decrease of the knot specific energy at high temperatures with increasing polymer lengths depending on whether the interactions are attractive or repulsive. Besides the investigation of the thermodynamics of polymer knots, the second goal of this paper is to introduce a method for distinguishing the topology of a knot based on a topological invariant which is in the form of multiple contour integrals and explicitly depends on the physical trajectory of the knot. The chosen invariant, denoted here ϱ(C), is related to the second coefficient of the Conway polynomial. It has been first isolated from the amplitudes of a Chern–Simons field theory with gauge group SU(N). It is shown that this invariant is very reliable in distinguishing the topology of polymer knots. One of the advantages of the proposed approach is that it allows one to reduce the number of samples needed by the Wang–Landau algorithm. Some solutions to speed up the calculations of ϱ(C) by exploiting Monte Carlo integration techniques are developed.

Correlators, Feynman Diagrams, and Quantum No-hair in deSitter Spacetime

We provide a parametric representation for position-space Feynman integrals of a massive, self-interacting scalar field in deSitter spacetime, for an arbitrary graph. The expression is given as a multiple contour integral over a kernel whose structure is determined by the set of all trees (or forests) within the graph, and it belongs to a class of generalized hypergeometric functions. We argue from this representation that connected deSitter n-point vacuum correlation functions have exponential decay for large proper time-separation, and also decay for large spatial separation, to arbitrary orders in perturbation theory. Our results may be viewed as an analog of the so-called cosmic-no-hair theorem in the context of a quantized test scalar field. This work has significant overlap with a paper by Marolf and Morrison, which is being released simultaneously.

Refined functional relations for the elliptic SOS model

In this work we refine the method presented in Galleas (2012) [1] and obtain a novel kind of functional equation determining the partition function of the elliptic SOS model with domain wall boundaries. This functional relation arises from the dynamical Yang–Baxter relation and its solution is given in terms of multiple contour integrals.

Multiple integral representation for the trigonometric SOS model with domain wall boundaries

Using the dynamical Yang–Baxter algebra we derive a functional equation for the partition function of the trigonometric SOS model with domain wall boundary conditions. The solution of the equation is given in terms of a multiple contour integral.

On recurrences for Ising integrals

We use WZ summation methods to compute recurrences for the Ising-class integrals C n , k . In this context, we describe an algorithmic approach to obtain homogeneous and inhomogeneous recurrences for a general class of multiple contour integrals of Barnes' type.

The nested SU(N) off-shell Bethe ansatz and exact form factors

The form factor equations are solved for a SU(N) invariant S-matrix under the assumption that the anti-particle is identified with the bound state of N − 1 particles. The solution is obtained explicitly in terms of the nested off-shell Bethe ansatz where the contribution from each level is written in terms of multiple contour integrals. The general solution is illustrated for some operators, such as the energy–momentum tensor, the fields and the current.

Another Look at the Performance of MRC Schemes in Nakagami-<tex>$m$</tex>Fading Channels With Arbitrary Parameters

In this letter, we take a close look at the performance of maximal ratio combining (MRC) schemes operating in a flat-Nakagami-m fading environment with arbitrary fading parameters. We derive an expression for the probability density function (pdf) of the output signal-to-noise ratio (SNR) by expressing the moment generating function of the output SNR in the form of multiple Barnes-type contour integrals. By evaluating the inverse transform and converting the multiple contour integrals into infinite series, we are able to derive an expression for the pdf of the output SNR when the Nakagami fading parameters along the diversity branches take on real and arbitrary values. Consequently, the average bit-error rate can now be expressed in terms of Lauricella's multivariate hypergeometric function, which can be easily evaluated numerically. Special cases of the main results reduce to known results in the literature. The results, which apply to independent as well as correlated diversity branches, will be useful for predicting the system performances when the Nakagami fading parameters are real and arbitrary.

Autocorrelation of Random Matrix Polynomials

We calculate the autocorrelation functions (or shifted moments) of the characteristic polynomials of matrices drawn uniformly with respect to Haar measure from the groups U(N), O(2N) and USp(2N). In each case the result can be expressed in three equivalent forms: as a determinant sum (and hence in terms of symmetric polynomials), as a combinatorial sum, and as a multiple contour integral. These formulae are analogous to those previously obtained for the Gaussian ensembles of Random Matrix Theory, but in this case are identities for any size of matrix, rather than large-matrix asymptotic approximations. They also mirror exactly autocorrelation formulae conjectured to hold for L-functions in a companion paper. This then provides further evidence in support of the connection between Random Matrix Theory and the theory of L-functions.

Fractional integration of the H-function of several variables

The main object of the present paper is to derive a number of key formulas for the fractional integration of the multivariable H -function (which is defined by a multiple contour integral of Mellin-Barnes type). Each of the general Eulerian integral formulas (obtained in this paper) are shown to yield interesting new results for various families of generalized hypergeometric functions of several variables. Some of these applications of the key formulas would provide potentially useful generalizations of known results in the theory of fractional calculus.

The convergence problem of certain multiple Mellin-Barnes contour integrals representing H-functions in several variables

Convergence regions for certain multiple Mellin-Barnes contour integrals representing the H-functions in several variables are determined. Many illustrative examples are also given.