34 research outputs found
Large deviations of spread measures for Gaussian matrices
For a large Gaussian matrix, we compute the joint statistics,
including large deviation tails, of generalized and total variance - the scaled
log-determinant and trace of the corresponding covariance
matrix. Using a Coulomb gas technique, we find that the Laplace transform of
their joint distribution decays for large (with
fixed) as , where is the Dyson index of the ensemble and
is a -independent large deviation function, which we compute exactly for
any . The corresponding large deviation functions in real space are worked
out and checked with extensive numerical simulations. The results are
complemented with a finite treatment based on the Laguerre-Selberg
integral. The statistics of atypically small log-determinants is shown to be
driven by the split-off of the smallest eigenvalue, leading to an abrupt change
in the large deviation speed.Comment: 20 pages, 3 figures. v4: final versio
Statistical distribution of the Wigner-Smith time-delay matrix moments for chaotic cavities
We derive the joint distribution of the moments
() of the Wigner-Smith matrix for a chaotic cavity supporting a
large number of scattering channels . This distribution turns out to be
asymptotically Gaussian, and we compute explicitly averages and covariances.
The results are in a compact form and have been verified numerically. The
general methodology of proof and computations has a wide range of applications.Comment: 5 pages, 1 table, 2 figure
Free fermions and the classical compact groups
There is a close connection between the ground state of non-interacting
fermions in a box with classical (absorbing, reflecting, and periodic) boundary
conditions and the eigenvalue statistics of the classical compact groups. The
associated determinantal point processes can be extended in two natural
directions: i) we consider the full family of admissible quantum boundary
conditions (i.e., self-adjoint extensions) for the Laplacian on a bounded
interval, and the corresponding projection correlation kernels; ii) we
construct the grand canonical extensions at finite temperature of the
projection kernels, interpolating from Poisson to random matrix eigenvalue
statistics. The scaling limits in the bulk and at the edges are studied in a
unified framework, and the question of universality is addressed. Whether the
finite temperature determinantal processes correspond to the eigenvalue
statistics of some matrix models is, a priori, not obvious. We complete the
picture by constructing a finite temperature extension of the Haar measure on
the classical compact groups. The eigenvalue statistics of the resulting grand
canonical matrix models (of random size) corresponds exactly to the grand
canonical measure of non-interacting free fermions with classical boundary
conditions.Comment: 35 pages, 5 figures. Final versio
Joint statistics of quantum transport in chaotic cavities
We study the joint statistics of conductance and shot noise in
chaotic cavities supporting a large number of open electronic channels in
the two attached leads. We determine the full phase diagram in the
plane, employing a Coulomb gas technique on the joint density of transmission
eigenvalues, as dictated by Random Matrix Theory. We find that in the region of
typical fluctuations, conductance and shot noise are uncorrelated and jointly
Gaussian, and away from it they fluctuate according to a different joint rate
function in each phase of the plane. Different functional forms of the
rate function in different regions emerge as a direct consequence of third
order phase transitions in the associated Coulomb gas problem.Comment: 6 pages, 1 figur
The oriented swap process and last passage percolation
We present new probabilistic and combinatorial identities relating three
random processes: the oriented swap process on particles, the corner growth
process, and the last passage percolation model. We prove one of the
probabilistic identities, relating a random vector of last passage percolation
times to its dual, using the duality between the Robinson-Schensted-Knuth and
Burge correspondences. A second probabilistic identity, relating those two
vectors to a vector of 'last swap times' in the oriented swap process, is
conjectural. We give a computer-assisted proof of this identity for
after first reformulating it as a purely combinatorial identity, and discuss
its relation to the Edelman-Greene correspondence. The conjectural identity
provides precise finite- and asymptotic predictions on the distribution of
the absorbing time of the oriented swap process, thus conditionally solving an
open problem posed by Angel, Holroyd and Romik.Comment: 36 pages, 6 figures. Full version of the FPSAC 2020 extended abstract
arXiv:2003.0333
Typical Entanglement
Let a pure state \psi be chosen randomly in an NM-dimensional Hilbert space,
and consider the reduced density matrix \rho of an N-dimensional subsystem. The
bipartite entanglement properties of \psi are encoded in the spectrum of \rho.
By means of a saddle point method and using a "Coulomb gas" model for the
eigenvalues, we obtain the typical spectrum of reduced density matrices. We
consider the cases of an unbiased ensemble of pure states and of a fixed value
of the purity. We finally obtain the eigenvalue distribution by using a
statistical mechanics approach based on the introduction of a partition
function.Comment: 15 pages, 4 figure
Universality of the third-order phase transition in the constrained Coulomb gas
The free energy at zero temperature of Coulomb gas systems in generic
dimension is considered as a function of a volume constraint. The transition
between the 'pulled' and the 'pushed' phases is characterised as a third-order
phase transition, in all dimensions and for a rather large class of isotropic
potentials. This suggests that the critical behaviour of the free energy at the
'pulled-to-pushed' transition may be universal, i.e., to some extent
independent of the dimension and the details of the pairwise interaction.Comment: 18 pages, 2 figures. v2: References adde
Universality of the weak pushed-to-pulled transition in systems with repulsive interactions
We consider a -dimensional gas in canonical equilibrium under pairwise
screened Coulomb repulsion and external confinement, and subject to a volume
constraint. We show that its excess free energy displays a generic third-order
singularity separating the pushed and pulled phases, irrespective of range of
the pairwise interaction, dimension and details of the confining potential. The
explicit expression of the excess free energy is universal and interpolates
between the Coulomb (long-range) and the delta (zero-range) interaction. The
order parameter of this transition - the electrostatic pressure generated by
the surface excess charge - is determined by invoking a fundamental energy
conservation argument.Comment: 12 pages, 2 figures. Revised versio
Random matrices associated to Young diagrams
We consider the singular values of certain Young diagram shaped random
matrices. For block-shaped random matrices, the empirical distribution of the
squares of the singular eigenvalues converges almost surely to a distribution
whose moments are a generalisation of the Catalan numbers. The limiting
distribution is the density of a product of rescaled independent Beta random
variables and its Stieltjes-Cauchy transform has a hypergeometric
representation. In special cases we recover the Marchenko-Pastur and
Dykema-Haagerup measures of square and triangular random matrices,
respectively. We find a further factorisation of the moments in terms of two
complex-valued random variables that generalises the factorisation of the
Marcenko-Pastur law as product of independent uniform and arcsine random
variables.Comment: 17 pages, 1 figur
The semiclassical limit of a quantum Zeno dynamics
Motivated by a quantum Zeno dynamics in a cavity quantum electrodynamics
setting, we study the asymptotics of a family of symbols corresponding to a
truncated momentum operator, in the semiclassical limit of vanishing Planck
constant and large quantum number , with kept
fixed. In a suitable topology, the limit is the discontinuous symbol
where is the characteristic function of the classically
permitted region in phase space. A refined analysis shows that the symbol
is asymptotically close to the function , where
is a smooth version of related to the integrated Airy
function. We also discuss the limit from a dynamical point of view.Comment: 28 pages, 5 figure