A representation of the moment measures of the general ideal Boe gas

We reconsider the fundamental work of Fichtner 2 and exhibit the permanental structure of the ideal Bose gas again, using a new approach which combines a characterization of infinitely divisible random measures (due to Kerstan, Kummer and Matthes 4 , 6 and Mecke 9 , 10 ) with a decomposition of the moment measures into its factorial measures due to Krickeberg 5 . To be more precise, we exhibit the moment measures of all orders of the general ideal Bose gas in terms of certain “loop” integrals. This representation can be considered as a point process analogue of the old idea of Symanzik 15 that local times and self‐crossings of the Brownian motion can be used as a tool in quantum field theory. Behind the notion of a general ideal Bose gas there is a class of infinitely divisible point processes of all orders with a Lévy‐measure belonging to some large class of measures containing that of the classical ideal Bose gas considered by Fichtner. It is well‐known that the calculation of moments of higher order of point processes is notoriously complicated. See for instance Krickeberg’s calculations for the Poisson or the Cox process in 5 . Relations to the work of Shirai, Takahashi 12 and Soshnikov 14 on permanental and determinantal processes are outlined.     

Asymptotics of Plancherel measures for the infinite-dimensional unitary group

We study a two-dimensional family of probability measures on infinite Gelfand-Tsetlin schemes induced by a distinguished family of extreme characters of the infinite-dimensional unitary group. These measures are unitary group analogs of the well-known Plancherel measures for symmetric groups.We show that any measure from our family defines a determinantal point process on Z₊×Z, and we prove that in appropriate scaling limits, such processes converge to two different extensions of the discrete sine process as well as to the extended Airy and Pearcey processes.     

Infinite-Dimensional Measure Spaces and Frame Analysis

We study certain infinite-dimensional probability measures in connection with frame analysis. Earlier work on frame-measures has so far focused on the case of finite-dimensional frames. We point out that there are good reasons for a sharp distinction between stochastic analysis involving frames in finite vs. infinite dimensions. For the case of infinite-dimensional Hilbert space ?, we study three cases of measures. We first show that, for ? infinite dimensional, one must resort to infinite dimensional measure spaces which properly contain ?. The three cases we consider are: (i) Gaussian frame measures, (ii) Markov path-space measures, and (iii) determinantal measures.     

A determinantal identity for skewfields

In [2] Bagazgoitia obtained a determinantal identity for the real quaternions. We show how his result may be regarded as one on reduced norms, and we proceed, in the same vein, to derive a determinantal identity for any skewfield which is finite- dimensional over its center.     

Averages of characteristic polynomials in random matrix theory

We compute averages of products and ratios of characteristic polynomials associated with orthogonal, unitary, and symplectic ensembles of random matrix theory. The Pfaffian/determinantal formulae for these averages are obtained, and the bulk scaling asymptotic limits are found for ensembles with Gaussian weights. Classical results for the correlation functions of the random matrix ensembles and their bulk scaling limits are deduced from these formulae by a simple computation. We employ a discrete approximation method: the problem is solved for discrete analogues of random matrix ensembles originating from representation theory, and then a limit transition is performed. Exact Pfaffian/determinantal formulae for the discrete averages are proven using standard tools of linear algebra; no application of orthogonal or skew‐orthogonal polynomials is needed. © 2005 Wiley Periodicals, Inc.     

Polynomials with and without determinantal representations

The problem of writing real zero polynomials as determinants of linear matrix polynomials has recently attracted a lot of attention. Helton and Vinnikov [9] have proved that any real zero polynomial in two variables has a determinantal representation. Brändén [2] has shown that the result does not extend to arbitrary numbers of variables, disproving the generalized Lax conjecture. We prove that in fact almost no real zero polynomial admits a determinantal representation; there are dimensional differences between the two sets. The result follows from a general upper bound on the size of linear matrix polynomials. We then provide a large class of surprisingly simple explicit real zero polynomials that do not have a determinantal representation. We finally characterize polynomials of which some power has a determinantal representation, in terms of an algebra with involution having a finite dimensional representation. We use the characterization to prove that any quadratic real zero polynomial has a determinantal representation, after taking a high enough power. Taking powers is thereby really necessary in general. The representations emerge explicitly, and we characterize them up to unitary equivalence.     

On the cotangent cohomology of rational surface singularities with almost reduced fundamental cycle

We prove dimension formulas for the cotangent spaces T ¹ and T ² for a class of rational surface singularities by calculating a correction term in the general dimension formulas. We get that it is zero if the dual graph of the rational surface singularity X does not contain a particular type of configurations, and this generalizes a result of Theo de Jong stating that the correction term c (X ) is zero for rational determinantal surface singularities. In particular our result implies that c (X ) is zero for Riemenschneiders quasi‐determinantal rational surface singularities, and this also generalizes results for quotient singularities. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A note on the Chow groups of projective determinantal varieties; an appendix to the paper by Grzegorz Kapustka and Michal Kapustka on “A cascade of determinantal Calabi‐Yau threefolds”

We investigate the Chow groups of projective determinantal varieties and those of their strata of matrices of fixed rank, using Chern class computations (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The local–global principle for symmetric determinantal representations of smooth plane curves

A smooth plane curve is said to admit a symmetric determinantal representation if it can be defined by the determinant of a symmetric matrix with entries in linear forms in three variables. We study the local–global principle for the existence of symmetric determinantal representations of smooth plane curves over a global field of characteristic different from two. When the degree of the plane curve is less than or equal to three, we relate the problem of finding symmetric determinantal representations to more familiar Diophantine problems on the Severi–Brauer varieties and mod 2 Galois representations, and prove that the local–global principle holds for conics and cubics. We also construct counterexamples to the local–global principle for quartics using the results of Mumford, Harris, and Shioda on theta characteristics.     

Critical behavior of nonintersecting Brownian motions at a tacnode

We study a model of n one‐dimensional, nonintersecting Brownian motions with two prescribed starting points at time t = 0 and two prescribed ending points at time t = 1 in a critical regime where the paths fill two tangent ellipses in the time‐space plane as n → ∞. The limiting mean density for the positions of the Brownian paths at the time of tangency consists of two touching semicircles, possibly of different sizes. We show that in an appropriate double scaling limit, there is a new family of limiting determinantal point processes with integrable correlation kernels that are expressed in terms of a new Riemann‐Hilbert problem of size 4 × 4. We prove solvability of the Riemann‐Hilbert problem and establish a remarkable connection with the Hastings‐McLeod solution of the Painlevé II equation. We show that this Painlevé II transcendent also appears in the critical limits of the recurrence coefficients of the multiple Hermite polynomials that are associated with the nonintersecting Brownian motions. Universality suggests that the new limiting kernels apply to more general situations whenever a limiting mean density vanishes according to two touching square roots, which represents a new universality class. © 2011 Wiley Periodicals, Inc     

On Goulden-Jackson’s Determinantal Formula for the Immanant

In 1992, Goulden and Jackson found a beautiful determinantal expression for the immanant of a matrix. This paper proves the same result combinatorially. We also present a β-extension of the theorem and a simple determinantal expression for the irreducible characters of the symmetric group.     

A Lower Bound for the Determinantal Complexity of a Hypersurface

We prove that the determinantal complexity of a hypersurface of degree \(d > 2\) is bounded below by one more than the codimension of the singular locus, provided that this codimension is at least 5. As a result, we obtain that the determinantal complexity of the \(3 \times 3\) permanent is 7. We also prove that for \(n> 3\), there is no nonsingular hypersurface in \({\mathbb {P}}^n\) of degree d that has an expression as a determinant of a \(d \times d\) matrix of linear forms, while on the other hand for \(n \le 3\), a general determinantal expression is nonsingular. Finally, we answer a question of Ressayre by showing that the determinantal complexity of the unique (singular) cubic surface containing a single line is 5.     

Determinants and current flows in electric networks

We consider electrical networks in which current enters at a single node and leaves at another. It has long been known that the currents and potential differences in such networks can be expressed in terms of determinants, or alternatively as counts of trees. Here we give alternative determinantal expressions.     

On indices of 1-forms on determinantal singularities

We consider 1-forms on so-called essentially isolated determinantal singularities (a natural generalization of isolated singularities), show how to define analogs of the Poincaré-Hopf index for them, and describe relations between these indices and the radial index. For isolated determinantal singularities, we discuss properties of the homological index of a holomorphic 1-form and its relation to the Poincaré-Hopf index.     

Mixed ladder determinantal varieties from two-sided ladders

We study the family of ideals defined by mixed size minors of two-sided ladders of indeterminates. We compute their Gröbner bases with respect to a skew-diagonal monomial order, then we use them to compute the height of the ideals. We show that these ideals correspond to a family of irreducible projective varieties, that we call mixed ladder determinantal varieties. We show that these varieties are arithmetically Cohen-Macaulay, and we characterize the arithmetically Gorenstein ones. Our main result consists in proving that mixed ladder determinantal varieties belong to the same G-biliaison class of a linear variety.     

Determinantal martingales and noncolliding diffusion processes

Two aspects of noncolliding diffusion processes have been extensively studied. One of them is the fact that they are realized as harmonic Doob transforms of absorbing particle systems in the Weyl chambers. Another aspect is integrability in the sense that any spatio-temporal correlation function can be expressed by a determinant. The purpose of the present paper is to clarify the connection between these two aspects. We introduce a notion of determinantal martingale and prove that, if the system has determinantal-martingale representation, then it is determinantal. In order to demonstrate the direct connection between the two aspects, we study three processes.     

Random partitions and the gamma kernel

We study the asymptotics of certain measures on partitions (the so-called z-measures and their relatives) in two different regimes: near the diagonal of the corresponding Young diagram and in the intermediate zone between the diagonal and the edge of the Young diagram. We prove that in both cases the limit correlation functions have determinantal form with a correlation kernel which depends on two real parameters. In the first case the correlation kernel is discrete, and it has a simple expression in terms of the gamma functions. In the second case the correlation kernel is continuous and translationally invariant, and it can be written as a ratio of two suitably scaled hyperbolic sines.     

Stationary Determinantal Processes on $${\mathbb {Z}}^d$$ Z d with N Labeled Objects per Site,Part I: Basic Properties and Full Domination

Journal of Theoretical Probability - We study a class of stationary determinantal processes on configurations of N labeled objects that may be present or absent at each site of $${\mathbb {Z}}^d$$...     

Noncolliding Jacobi processes as limits of Markov chains on the Gelfand–Tsetlin graph

We introduce a stochastic dynamics related to the measures that arise in harmonic analysis on the infinite–dimensional unitary group. Our dynamics is obtained as a limit of a sequence of natural Markov chains on the Gelfand–Tsetlin graph. We compute the finite-dimensional distributions of the limit Markov process, the generator and eigenfunctions of the semigroup related to this process. The limit process can be identified with the Doob h-transform of a family of independent diffusions. The space-time correlation functions of the limit process have a determinantal form. Bibliography: 21 titles. Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 360, 2008, pp. 91–123.