195 research outputs found
Intertwinings for general β -Laguerre and β -Jacobi processes
No full textWe show that, for β≥1, the semigroups of β-Laguerre and β-Jacobi processes of different dimensions are intertwined in analogy to a similar result for β-Dyson Brownian motion recently obtained in Ramanan and Shkolnikov (Intertwinings of β-Dyson Brownian motions of different dimensions, 2016. arXiv:1608.01597). These intertwining relations generalize to arbitrary β≥1 the ones obtained for β=2 in Assiotis et al. (Interlacing diffusions, 2016. arXiv:1607.07182) between h-transformed Karlin–McGregor semigroups. Moreover, they form the key step toward constructing a multilevel process in a Gelfand–Tsetlin pattern leaving certain Gibbs measures invariant. Finally, as a by-product, we obtain a relation between general β-Jacobi ensembles of different dimensions
Exact Solution of Interacting Particle Systems Related to Random Matrices
No full textWe consider one-dimensional diffusions, with polynomial drift and diffusion coefficients, so that in particular themotion can be space-inhomogeneous, interacting via one-sided reflections. The prototypical example is the well-known model of Brownian motions with one-sided collisions, also known as Brownian TASEP, which is equivalent to Brownian last passage percolation. We obtain a formula for the finite dimensional distributions of these particle systems, starting from arbitrary initial condition, in terms of a Fredholm determinant of an explicit kernel. As far as we can tell, in the spaceinhomogeneous setting and for general initial condition this is the first time such a result has been proven.We moreover consider themodel of non-colliding diffusions, again with polynomial drift and diffusion coefficients, which includes the ones associated to all the classical ensembles of random matrices. We prove that starting from arbitrary initial condition the induced point process has determinantal correlation functions in space and time with an explicit correlation kernel. A key ingredient in our general method of exact solution for both models is the application of the backward in time diffusion flow on certain families of polynomials constructed from the initial condition
A matrix Bougerol identity and the Hua-Pickrell measures
No full textWe prove a Hermitian matrix version of Bougerol's identity. Moreover, we construct the Hua-Pickrell measures on Hermitian matrices, as stochastic integrals with respect to a drifting Hermitian Brownian motion and with an integrand involving a conjugation by an independent, matrix analogue of the exponential of a complex Brownian motion with drift
Ergodic decomposition for inverse Wishart measures on infinite positive-definite matrices
No full textThe ergodic unitarily invariant measures on the space of infinite Hermitian matrices have been classified by Pickrell and Olshanski-Vershik. The much-studied complex inverse Wishart measures form a projective family, thus giving rise to a unitarily invariant measure on infinite positive-definite matrices. In this paper we completely solve the corresponding problem of ergodic decomposition for this measure
Random entire functions from random polynomials with real zeros
No full textWe point out a simple criterion for convergence of polynomials to a concrete
entire function in the Laguerre-P\'{o}lya () class (of all
functions arising as uniform limits of polynomials with only real roots). We
then use this to show that any random function can be obtained
as the uniform limit of rescaled characteristic polynomials of principal
submatrices of an infinite unitarily invariant random Hermitian matrix.
Conversely, the rescaled characteristic polynomials of principal submatrices of
any infinite random unitarily invariant Hermitian matrix converge uniformly to
a random function. This result also has a natural extension to
-ensembles. Distinguished cases include random entire functions
associated to the -Sine, and more generally -Hua-Pickrell,
-Bessel and -Airy point processes studied in the literature.Comment: Improvements following referee report. To appear Advances in Mat
On the moments of the partition function of the CβE field
No full textWe obtain a combinatorial formula for the positive integer moments of the partition function of the CβEN field, or equivalently the moments of the moments of the characteristic polynomial of the CβEN ensemble. We then use this formula to establish the large N asymptotics of these moments in the "moment-supercritical" regime. A key role is played by Jack polynomials
Random surface growth and Karlin-McGregor polynomials
No full textWe consider consistent dynamics for non-intersecting birth and death chains, originating from dualities of stochastic coalescing flows and one dimensional orthogonal polynomials. As corollaries, we obtain unified and simple probabilistic proofs of certain key intertwining relations between multivariate Markov chains on the levels of some branching graphs. Special cases include the dynamics on the Gelfand-Tsetlin graph considered by Borodin and Olshanski and the ones on the BC-type graph recently studied by Cuenca. Moreover, we introduce a general inhomogeneous random growth process with a wall that includes as special cases the ones considered by Borodin and Kuan and Cerenzia, that are related to the representation theory of classical groups and also the Jacobi growth process more recently studied by Cerenzia and Kuan. Its most important feature is that, this process retains the determinantal structure of the ones studied previously and for the fully packed initial condition we are able to calculate its correlation kernel explicitly in terms of a contour integral involving orthogonal polynomials. At a certain scaling limit, at a finite distance from the wall, one obtains for a single level discrete determinantal ensembles associated to continuous orthogonal polynomials, that were recently introduced by Borodin and Olshanski, and that depend on the inhomogeneities
Exact solution of interacting particle systems related to random matrices
No full textWe consider one-dimensional diffusions, with polynomial drift and diffusion
coefficients, so that in particular the motion can be space-inhomogeneous,
interacting via one-sided reflections. The prototypical example is the
well-known model of Brownian motions with one-sided collisions, also known as
Brownian TASEP, which is equivalent to Brownian last passage percolation. We
obtain a formula for the finite dimensional distributions of these particle
systems, starting from arbitrary initial condition, in terms of a Fredholm
determinant of an explicit kernel. As far as we can tell, in the
space-inhomogeneous setting and for general initial condition this is the first
time such a result has been proven. We moreover consider the model of
non-colliding diffusions, again with polynomial drift and diffusion
coefficients, which includes the ones associated to all the classical ensembles
of random matrices. We prove that starting from arbitrary initial condition the
induced point process has determinantal correlation functions in space and time
with an explicit correlation kernel. A key ingredient in our general method of
exact solution for both models is the application of the backward in time
diffusion flow on certain families of polynomials constructed from the initial
condition.Comment: Revised following referee reports. To appear CM
A kinetic approach for the estimation of intracellular concentrations of nitrosative species in cells challenged by nitric oxide
No full textThesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Chemical Engineering, 2013.This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.Cataloged from student-submitted PDF version of thesis.Includes bibliographical references.by Vasileios Theodoros Dendroulakis.Ph.D
On the singular values of complex matrix Brownian motion with a matrix drift
No full textLet MatC(K,N) be the space of K×N complex matrices. Let Bt be Brownian motion on MatC(K,N) starting from the zero matrix and M∈MatC(K,N). We prove that, with K≥N, the N eigenvalues of (Bt+tM)∗(Bt+tM) form a Markov process with an explicit transition kernel. This generalizes a classical result of Rogers and Pitman for multidimensional Brownian motion with drift which corresponds to N=1. We then give two more descriptions for this Markov process. First, as independent squared Bessel diffusion processes in the wide sense, introduced by Watanabe and studied by Pitman and Yor, conditioned to never intersect. Second, as the distribution of the top row of interacting squared Bessel type diffusions in some interlacting array. The last two descriptions also extend to a general class of one-dimensional diffusions
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