Dyson Integral Research Articles

Power spectra and autocovariances of level spacings beyond the Dyson conjecture.

Introduced in the early days of random matrix theory, the autocovariances δI_{k}^{j}=cov(s_{j},s_{j+k}) of level spacings {s_{j}} accommodate detailed information on the correlations between individual eigenlevels. It was first conjectured by Dyson that the autocovariances of distant eigenlevels in the unfolded spectra of infinite-dimensional random matrices should exhibit a power-law decay δI_{k}^{j}≈-1/βπ^{2}k^{2}, where β is the symmetry index. In this Letter, we establish an exact link between the autocovariances of level spacings and their power spectrum, and show that, for β=2, the latter admits a representation in terms of a fifth Painlevé transcendent. This result is further exploited to determine an asymptotic expansion for autocovariances that reproduces the Dyson formula as well as provides the subleading corrections to it. High-precision numerical simulations lend independent support to our results.

Application of the Pick Function in the Lieb Concavity Theorem for Deformed Exponentials

The Lieb concavity theorem, successfully solved in the Wigner–Yanase–Dyson conjecture, is an important application of matrix concave functions. Recently, the Thompson–Golden theorem, a corollary of the Lieb concavity theorem, was extended to deformed exponentials. Hence, it is worthwhile to study the Lieb concavity theorem for deformed exponentials. In this paper, the Pick function is used to obtain a generalization of the Lieb concavity theorem for deformed exponentials, and some corollaries associated with exterior algebra are obtained.

Critical points, critical values, and a determinant identity for complex polynomials

Given any n-tuple of complex numbers, one can canonically define a polynomial of degree n+1 that has the entries of this n-tuple as its critical points. In 2002, Beardon, Carne, and Ng studied a map $\theta\colon \mathbb{C}^n\to \mathbb{C}^n$ which outputs the critical values of the canonical polynomial constructed from the input, and they proved that this map is onto. Along the way, they showed that $\theta$ is a local homeomorphism whenever the entries of the input are distinct and nonzero, and, implicitly, they produced a polynomial expression for the Jacobian determinant of $\theta$. In this article we extend and generalize both the local homeomorphism result and the elegant determinant identity to analogous situations where the critical points occur with multiplicities. This involves stratifying $\mathbb{C}^n$ according to which coordinates are equal and generalizing $\theta$ to a similar map $\mathbb{C}^\ell \to \mathbb{C}^\ell$ where $\ell$ is the number of distinct critical points. The more complicated determinant identity that we establish is closely connected to the multinomial identity known as Dyson's conjecture.

Constant Terms of Near-Dyson Polynomials

We formulate and prove a formula for the constant term for a certain class of Laurent polynomials, which include the Dyson conjecture and its generalizations by Bressoud and Goulden. Our method is explicit Combinatorial Nullstellensatz.

Transformation properties for Dyson’s rank function

At the 1987 Ramanujan Centenary meeting Dyson asked for a coherent group-theoretical structure for Ramanujan's mock theta functions analogous to Hecke's theory of modular forms. Many of Ramanujan's mock theta functions can be written in terms of $R(\zeta,q)$, where $R(z,q)$ is the two-variable generating function of Dyson's rank function and $\zeta$ is a root of unity. Building on earlier work of Watson, Zwegers, Gordon and McIntosh, and motivated by Dyson's question, Bringmann, Ono and Rhoades studied transformation properties of $R(\zeta,q)$. In this paper we strengthen and extend the results of Bringmann, Rhoades and Ono, and the later work of Ahlgren and Treneer. As an application we give a new proof of Dyson's rank conjecture and show that Ramanujan's Dyson rank identity modulo $5$ from the Lost Notebook has an analogue for all primes greater than $3$. The proof of this analogue was inspired by recent work of Jennings-Shaffer on overpartition rank differences mod $7$.

Compact Lie groups: Euler constructions and generalized Dyson conjecture

A generalized Euler parameterization of a compact Lie group is a way for parameterizing the group starting from a maximal Lie subgroup, which allows a simple characterization of the range of parameters. In the present paper we consider the class of all compact connected Lie groups. We present a general method for realizing their generalized Euler parameterization starting from any symmetrically embedded Lie group. Our construction is based on a detailed analysis of the geometry of these groups. As a byproduct this gives rise to an interesting connection with certain Dyson integrals. In particular, we obtain a geometry based proof of a Macdonald conjecture regarding the Dyson integrals correspondent to the root systems associated to all irreducible symmetric spaces. As an application of our general method we explicitly parameterize all groups of the class of simple, simply connected compact Lie groups. We provide a table giving all necessary ingredients for all such Euler parameterizations.

Spin gravitational resonance and graviton detection

We develop a gravitational analogue of spin magnetic resonance, called spin gravitational resonance, whereby a gravitational wave interacts with a magnetic field to produce a spin transition. In particular, an external magnetic field separates the energy spin states of a spin-1/2 particle, and the presence of the gravitational wave produces a perturbation in the components of the magnetic field orthogonal to the gravitational wave propagation. In this framework we test Dyson's conjecture that individual gravitons cannot be detected. Although we find no fundamental laws preventing single gravitons being detected with spin gravitational resonance, we show that it cannot be used in practice, in support of Dyson's conjecture.

Feasible Schedules for Rotating Transmissions

Motivated by a scheduling problem that arises in the study of optical networks, we prove the following result, which is a variation of a conjecture of Haxell, Wilfong and Winkler.Let , where the endpoints are reduced modulo 1, are pairwise disjoint on the unit circle.The proof is based on some properties of multivariate polynomials and on the validity of the Dyson conjecture.

A proof of Andrews’ q-Dyson conjecture

Let ( y ) a = ( 1 - y ) ( 1 - qy ) ⋯ ( 1 - q a - 1 y ) . We prove that the constant term of the Laurent polynomial ∏ 1 ⩽ i < j ⩽ n ( x i / x j ) a i ( qx j / x i ) a j , where x 1 , … , x n , q are commuting indeterminates and a 1 , … , a n are non-negative integers, equals ( q ) a 1 + ⋯ + a n / ( q ) a 1 … ( q ) a n . This settles in the affirmative a conjecture of George Andrews (in: R.A. Askey, ed., Theory and Applications of Special Functions, Academic Press, New York, 1975, 191–224].

A short proof of the Zeilberger-Bressoud $q$-Dyson theorem

We give a formal Laurent series proof of Andrews's q-Dyson Conjecture, first proved by Zeilberger and Bressoud.

Disturbing the Dyson conjecture, in a generally GOOD way

Dyson's celebrated constant term conjecture [F.J. Dyson, Statistical theory of the energy levels of complex systems I, J. Math. Phys. 3 (1962) 140–156] states that the constant term in the expansion of ∏ 1 ≦ i ≠ j ≦ n ( 1 − x i / x j ) a j is the multinomial coefficient ( a 1 + a 2 + ⋯ + a n ) ! / ( a 1 ! a 2 ! ⋯ a n ! ) . The definitive proof was given by I.J. Good [I.J. Good, Short proof of a conjecture of Dyson, J. Math. Phys. 11 (1970) 1884]. Later, Andrews extended Dyson's conjecture to a q-analog [G.E. Andrews, Problems and prospects for basic hypergeometric functions, in: R. Askey (Ed.), The Theory and Application of Special Functions, Academic Press, New York, 1975, pp. 191–224]. In this paper, closed form expressions are given for the coefficients of several other terms in the Dyson product, and are proved using an extension of Good's idea. Also, conjectures for the corresponding q-analogs are supplied. Finally, perturbed versions of the q-Dixon summation formula are presented.

A residue theorem for Malcev–Neumann series

In this paper, we establish a residue theorem for Malcev–Neumann series that requires few constraints, and includes previously known combinatorial residue theorems as special cases. Our residue theorem identifies the residues of two formal series (over a field of characteristic zero) which are related by a change of variables. We obtain simple conditions for when a change of variables is possible, and find that the two related formal series in fact belong to two different fields of Malcev–Neumann series. The multivariate Lagrange inversion formula is easily derived and Dyson's conjecture is given a new proof and generalized.

Ground State Energy of the Two-Component Charged Bose Gas

We continue the study of the two-component charged Bose gas initiated by Dyson in 1967. He showed that the ground state energy for $N$ particles is at least as negative as $-CN^{7/5}$ for large $N$ and this power law was verified by a lower bound found by Conlon, Lieb and Yau in 1988. Dyson conjectured that the exact constant $C$ was given by a mean-field minimization problem that used, as input, Foldy's calculation (using Bogolubov's 1947 formalism) for the one-component gas. Earlier we showed that Foldy's calculation is exact insofar as a lower bound of his form was obtained. In this paper we do the same thing for Dyson's conjecture. The two-component case is considerably more difficult because the gas is very non-homogeneous in its ground state.

Constant term identities and hypergeometric functions on spaces of Hermitian matrices

We announce the generalization, to the setting of hypergeometric functions of matrix argument, of classical constant term identities. As special cases, we obtain some constant term identities which play a prominent role in the proof of Dyson's conjecture.

Divided Differences and Combinatorial Identities

We present an algebraic theory of divided differences which includes confluent differences, interpolation formulas, Liebniz's rule, the chain rule, and Lagrange inversion. Our approach uses only basic linear algebra. We also show that the general results about divided differences yield interesting combinatorial identities when we consider some suitable particular cases. For example, the chain rule gives us generalizations of the identity used by Good in his famous proof of Dyson's conjecture. We also obtain identities involving binomial coefficients, Stirling numbers, Gaussian coefficients, and harmonic numbers.

Problems on the Z-statistic

We present ten assorted problems which have arisen in the attempt to understand the Z-statistic, a statistic on words which plays a crucial role in the proof of Andrews' q-Dyson conjecture.

The extension to root systems of a theorem on tournaments

M. G. Kendall and B. Babington-Smith proved that if a tournament p′ is obtained from a tournament p by reversing the edges of a 3-cycle then p and p′ contain the same number of 3-cycles. This theorem is the basis of a cancellation argument used by D. Zeilberer and D. M. Bressoud in their recent proof of the q-analog of Dyson's conjecture. The theorem may be restated in terms of the root system A n and the main result of this paper is the extension of this theorem to arbitrary root systems. As one application we give a combinatorial proof of a special case of the Macdonald conjecture for root systems using the method of Zeilberger and Bressoud. A second application is a combinatorial proof of the Weyl denominator formula.

A Proof of Andrews' q-Dyson Conjecture for n = 4

A Proof of Andrews' q-Dyson Conjecture for n = 4

A proof of Andrews' q-Dyson conjecture

Let ( y) a =(1− y)(1− qy)…(1− q a−1 y). We prove that the constant term of the Laurent polynomial Π 1⩽ i< j⩽ n ( x i / x j ) a i ( qx j / x i ) a j , where x 1,…, x n , q are commmuting indeterminates and a 1,…, a n are non-negative integers, equals ( q) a 1+… + a n /( q) a n . This settles in the affirmative a conjecture of George Andrews (in: R.A. Askey, ed., Theory and Applications of Special Functions, Academic Press, New York, 1975, 191–224].

A proof of Andrews’ $q$-Dyson conjecture for $n=4$

Andrews’ $q$-Dyson conjecture is that the constant term in a polynomial associated with the root system ${A_{n - 1}}$ is equal to the $q$-multinomial coefficient. Good used an identity to establish the case $q = 1$, which was originally raised by Dyson. Andrews established his conjecture for $n \leqslant 3$ and Macdonald proved it when ${a_1} = {a_2} = \cdots = {a_n} = 1,2$ or $\infty$ for all $n \geqslant 2$. We use a $q$-analog of Good’s identity which involves a remainder term and linear algebra to establish the conjecture for $n = 4$. The remainder term arises because of an essential problem with the $q$-Dyson conjecture: the symmetry of the constant term. We give a number of conjectures related to the symmetry.