Cauchy Determinant Research Articles

Of shadows and gaps in spatial search

Spatial search occurs in a connected graph if a continuous-time quantum walk on the adjacency matrix of the graph, suitably scaled, plus a rank-one perturbation induced by any vertex will unitarily map the principal eigenvector of the graph to the characteristic vector of the vertex. This phenomenon is a natural continuous-time analogue of Grover search. The spatial search is said to be optimal if it occurs with constant fidelity and in time inversely proportional to the shadow of the target vertex on the principal eigenvector. Extending a result of Chakraborty \etal ({\em Physical Review A}, {\bf 102}:032214, 2020), we prove a simpler characterization of optimal spatial search. Based on this characterization, we observe that some families of distance-regular graphs, such as Hamming and Grassmann graphs, have optimal spatial search. We also show a matching lower bound on time for spatial search with constant fidelity, which extends a bound due to Farhi and Gutmann for perfect fidelity. Our elementary proofs employ standard tools, such as Weyl inequalities and Cauchy determinant formula.

On a generalized Weierstrass theta identity

The classical Weierstrass theta identity plays a central role in the study of modular, elliptic and theta hypergeometric series. In this paper, we establish a multi-parameter Weierstrass theta identity with the use of the Cauchy determinant and an anti-symmetric difference of a product of two theta functions. Some applications of this generalized Weierstrass theta identity to basic and elliptic hypergeometric series are also discussed.

New solutions to the differential-difference KP equation

In this paper, we study the differential-difference Kadomtsev–Petviashvili (DΔKP) equation through the dressing method based on a nonlocal ∂̄-problem. Cauchy type determinant solution and a new type rational solution are given. The initial value problem of DΔKP is discussed by virtue of the quasi-local ∂̄-problem. From an impulse initial condition, a new explicit solution to the initial value problem of DΔKP is obtained.

The q-TASEP with a Random Initial Condition

When studying fluctuations of models in the 1D KPZ class including the ASEP and the $q$-TASEP, a standard approach has been to first write down a formula for $q$-deformed moments and constitute their generating function. This works well for the step initial condition, but there is a difficulty for a random initial condition (including the stationary case): in this case only the first few moments are finite and the rest diverge. In a previous work [16], we presented a method dealing directly with the $q$-deformed Laplace transform of an observable, in which the above difficulty does not appear. There the Ramanujan's summation formula and the Cauchy determinant for the theta functions play an important role. In this note, we give an alternative approach for the $q$-TASEP without using them.

Fluctuations for stationary q-TASEP

We consider the q-totally asymmetric simple exclusion process (q-TASEP) in the stationary regime and study the fluctuation of the position of a particle. We first observe that the problem can be studied as a limiting case of an N-particle q-TASEP with a random initial condition and with particle dependent hopping rate. Then we explain how this N-particle q-TASEP can be encoded in a dynamics on a two-sided Gelfand–Tsetlin cone described by a two-sided q-Whittaker process and present a Fredholm determinant formula for the q-Laplace transform of the position of a particle. Two main ingredients in its derivation is the Ramanujan’s bilateral summation formula and the Cauchy determinant identity for the theta function with an extra parameter. Based on this we establish that the position of a particle obeys the universal stationary KPZ distribution (the Baik–Rains distribution) in the long time limit.

Determinant representations of spin-operator matrix elements in the XX spin chain and their applications

For the one-dimensional spin-1/2 XX model with either periodic or open boundary conditions, it is shown by using a fermionic approach that the matrix element of the spin operator $S^-_j$ ($S^-_{j}S^+_{j'}$) between two eigenstates with numbers of excitations $n$ and $n+1$ ($n$ and $n$) can be expressed as the determinant of an appropriate $(n+1)\times (n+1)$ matrix whose entries involve the coefficients of the canonical transformations diagonalizing the model. In the special case of a homogeneous periodic XX chain, the matrix element of $S^-_j$ reduces to a variant of the Cauchy determinant that can be evaluated analytically to yield a factorized expression. The obtained compact representations of these matrix elements are then applied to two physical scenarios: (i) Nonlinear optical response of molecular aggregates, for which the determinant representation of the transition dipole matrix elements between eigenstates provides a convenient way to calculate the third-order nonlinear responses for aggregates from small to large sizes compared with the optical wavelength, and (ii) real-time dynamics of an interacting Dicke model consisting of a single bosonic mode coupled to a one-dimensional XX spin bath. In this setup, full quantum calculation up to $N\leq 16$ spins for vanishing intrabath coupling shows that the decay of the reduced bosonic occupation number approaches a finite plateau value (in the long-time limit) that depends on the ratio between the number of excitations and the total number of spins. Our results can find useful applications in various "system-bath" systems, with the system part inhomogeneously coupled to an interacting XX chain.

Moments of the Hermitian Matrix Jacobi Process

In this paper, we compute the expectation of traces of powers of the Hermitian matrix Jacobi process for a large enough but fixed size. To proceed, we first derive the semi-group density of its eigenvalues process as a bilinear series of symmetric Jacobi polynomials. Next, we use the expansion of power sums in the Schur polynomial basis and the integral Cauchy–Binet formula in order to determine the partitions having nonzero contributions after integration. It turns out that these are hooks of bounded weight and the sought expectation results from the integral of a product of two Schur functions with respect to a generalized beta distribution. For special values of the parameters on which the matrix Jacobi process depends, the last integral reduces to the Cauchy determinant and we close the paper with the investigation of the asymptotic behavior of the resulting formula as the matrix size tends to infinity.

Three-sphere free energy for classical gauge groups

In this note, we calculate the S 3 free energy F of 3-d $ \mathcal{N} $ ≥ 4 supersymmetric gauge theories with U(N), O(N), and USp(2N) gauge groups and matter hypermultiplets in the fundamental and two-index tensor representations. Supersymmetric localization reduces the computation of F to a matrix model that we solve in the large N limit using two different methods. The first method is a saddle point approximation first introduced in [1], which we extend to next-to-leading order in 1/N. The second method generalizes the Fermi gas approach of [2] to theories with symplectic and orthogonal gauge groups, and yields an expression for F valid to all orders in 1/N . In developing the second method, we use a non-trivial generalization of the Cauchy determinant formula.

Direct derivation of “mirror” ABJ partition function

We study the partition function of the three-dimensional N=6 U(N)_k x U(N+M)_{-k} superconformal Chern-Simons matter theory known as the ABJ theory. We prove that the ABJ partition function on S^3 is exactly the same as a formula recently proposed by Awata, Hirano and Shigemori. While this formula was previously obtained by an analytic continuation from the L(2,1) lens space matrix model, we directly derive this by using a generalization of the Cauchy determinant identity. We also give an interpretation for the formula from brane picture.

Three dimensional mirror symmetry and partition function on S 3

We provide non-trivial checks of $ \mathcal{N} $ = 4, D = 3 mirror symmetry in a large class of quiver gauge theories whose Type IIB (Hanany-Witten) descriptions involve D3 branes ending on orbifold/orientifold 5-planes at the boundary. From the M-theory perspective, such theories can be understood in terms of coincident M2 branes sitting at the origin of a product of an A-type and a D-type ALE (Asymtotically Locally Euclidean) space with G-fluxes. Families of mirror dual pairs, which arise in this fashion, can be labeled as (A m−1 , D n ), where m and n are integers. For a large subset of such infinite families of dual theories, corresponding to generic values of n ≥ 4, arbitrary ranks of the gauge groups and varying m, we test the conjectured duality by proving the precise equality of the S 3 partition functions for dual gauge theories in the IR as functions of masses and FI parameters. The mirror map for a given pair of mirror dual theories can be read off at the end of this computation and we explicitly present these for the aforementioned examples. The computation uses non-trivial identities of hyperbolic functions including certain generalizations of Cauchy determinant identity and Schur’s Pfaffian identity, which are discussed in the paper.

Divided differences and a unified approach to evaluating determinants involving generalized factorials

By employing divided differences, a unified approach to the evaluation of some determinant involving generalized factorials is proposed. Previous generalizations of the Vandermonde determinant and the Cauchy determinant due to Chu-Claudio, Chu-Wang-Zhang and Johnson are included as special cases of our unified treatment.

Spontaneous magnetization of the superintegrable chiral Potts model: calculation of the determinant DPQ

For the Ising model, the calculation of the spontaneous magnetization leads to the problem of evaluating a determinant. Yang did this by calculating the eigenvalues in the large-lattice limit. Montroll, Potts and Ward expressed it as a Toeplitz determinant and used Szegő's theorem: this is almost certainly the route originally travelled by Onsager. For the corresponding problem in the superintegrable chiral Potts model, neither approach appears to work: here we show that the determinant DPQ can be expressed as that of a product of two Cauchy-like matrices. One can then use the elementary exact formula for the Cauchy determinant. One of course regains the known result, originally conjectured in 1989.

Generalizations of Cauchy's determinant and Schur's Pfaffian

We present several identities of Cauchy-type determinants and Schur-type Pfaffians involving generalized Vandermonde determinants, which generalize Cauchy's determinant det ( 1 / ( x i + y j ) ) and Schur's Pfaffian Pf ( ( x j − x i ) / ( x j + x i ) ) . Some special cases of these identities are given by S. Okada and T. Sundquist. As an application, we give a relation for the Littlewood–Richardson coefficients involving a rectangular partition.

An elliptic generalization of Schur's Pfaffian identity

We present a Pfaffian identity involving elliptic functions, whose rational limit gives a generalization of Schur's Pfaffian identity for Pf((xj-xi)/(xj+xi)). This identity can be regarded as the Pfaffian counterpart of Frobenius’ generalization of Cauchy's determinant identity for det(1/(xi+xj)).

The Flagged Cauchy Determinant

We consider a flagged form of the Cauchy determinant, for which we provide a combinatorial interpretation in terms of nonintersecting lattice paths. In combination with the standard determinant for the enumeration of nonintersecting lattice paths, we are able to give a new proof of the Cauchy identity for Schur functions. Moreover, by choosing different starting and end points for the lattice paths, we are led to a lattice path proof of an identity of Gessel which expresses a Cauchy-like sum of Schur functions in terms of the complete symmetric functions.

Formfactors and the functional form of correlators in the XX-spin chain

We present new expressions for the formfactors of local operators for the XX-quantum spinchain as Cauchy determinants. Using the known functional form of the correlator at largedistances we propose a new expression for the constant for the asymptotics of thecorrelator as a Cauchy determinant. We calculate the momentum distribution for thegeneral case of the XXZ-spin chain and point out that it is completely differentfrom that for the Luttinger model (the system of fermions). For the XX chain wecompare numerically the value of the lowest formfactor and the expectation value ofmomentum-zero operators which is determined by the functional form of the correlator.

Multiple elliptic hypergeometric series. An approach from the Cauchy determinant

A multiple generalization of elliptic hypergeometric series is studied through the Cauchy determinant for the Weierstrass sigma function. In particular, a duality transformation for multiple hypergeometric series is proposed. As an application, two types of Bailey transformations for very well-poised multiple elliptic hypergeometric series are derived.

Generalizations of the Cauchy determinant

Generalizations of the Cauchy determinant

A Determinant of the Chudnovskys Generalizing the Elliptic Frobenius-Stickelberger-Cauchy Determinantal Identity

D.V. Chudnovsky and G.V. Chudnovsky [CH] introduced a generalization of the Frobenius-Stickelberger determinantal identity involving elliptic functions that generalize the Cauchy determinant. The purpose of this note is to provide a simple essentially non-analytic proof of this evaluation. This method of proof is inspired by D. Zeilberger's creative application in [Z1].

Darboux Transformations for a Lax Integrable System in 2n Dimensions

A $2n$-dimensional Lax integrable system is proposed by a set of specific spectral problems. It contains Takasaki equations, the self-dual Yang-Mills equations and its integrable hierarchy as examples. An explicit formulation of Darboux transformations is established for this Lax integrable system. The Vandermonde and generalized Cauchy determinant formulas lead to a description for deriving explicit solutions and thus some rational and analytic solutions are obtained.