18 research outputs found
Multiple Meixner-Pollaczek polynomials and the six-vertex model
We study multiple orthogonal polynomials of Meixner-Pollaczek type with
respect to a symmetric system of two orthogonality measures. Our main result is
that the limiting distribution of the zeros of these polynomials is one
component of the solution to a constrained vector equilibrium problem. We also
provide a Rodrigues formula and closed expressions for the recurrence
coefficients. The proof of the main result follows from a connection with the
eigenvalues of block Toeplitz matrices, for which we provide some general
results of independent interest.
The motivation for this paper is the study of a model in statistical
mechanics, the so-called six-vertex model with domain wall boundary conditions,
in a particular regime known as the free fermion line. We show how the multiple
Meixner-Pollaczek polynomials arise in an inhomogeneous version of this model.Comment: 32 pages, 4 figures. References adde
What is a multiple orthogonal polynomial?
This is an extended version of our note in the Notices of the American
Mathematical Society 63 (2016), no. 9, in which we explain what multiple
orthogonal polynomials are and where they appear in various applications.Comment: 5 pages, 2 figure
Average Characteristic Polynomials of Determinantal Point Processes
We investigate the average characteristic polynomial where the 's are real random variables
which form a determinantal point process associated to a bounded projection
operator. For a subclass of point processes, which contains Orthogonal
Polynomial Ensembles and Multiple Orthogonal Polynomial Ensembles, we provide a
sufficient condition for its limiting zero distribution to match with the
limiting distribution of the random variables, almost surely, as goes to
infinity. Moreover, such a condition turns out to be sufficient to strengthen
the mean convergence to the almost sure one for the moments of the empirical
measure associated to the determinantal point process, a fact of independent
interest. As an application, we obtain from a theorem of Kuijlaars and Van
Assche a unified way to describe the almost sure convergence for classical
Orthogonal Polynomial Ensembles. As another application, we obtain from
Voiculescu's theorems the limiting zero distribution for multiple Hermite and
multiple Laguerre polynomials, expressed in terms of free convolutions of
classical distributions with atomic measures.Comment: 26 page
Return probability after a quench from a domain wall initial state in the spin-1/2 XXZ chain
We study the return probability and its imaginary () time continuation
after a quench from a domain wall initial state in the XXZ spin chain, focusing
mainly on the region with anisotropy . We establish exact Fredholm
determinant formulas for those, by exploiting a connection to the six vertex
model with domain wall boundary conditions. In imaginary time, we find the
expected scaling for a partition function of a statistical mechanical model of
area proportional to , which reflects the fact that the model exhibits
the limit shape phenomenon. In real time, we observe that in the region
the decay for large times is nowhere continuous as a function
of anisotropy: it is either gaussian at root of unity or exponential otherwise.
As an aside, we also determine that the front moves as , by analytic continuation of known arctic curves in
the six vertex model. Exactly at , we find the return probability
decays as . It is argued that this
result provides an upper bound on spin transport. In particular, it suggests
that transport should be diffusive at the isotropic point for this quench.Comment: 33 pages, 8 figures. v2: typos fixed, references added. v3: minor
change
Exact time evolution formulae in the XXZ spin chain with domain wall initial state
We study the time evolution of the spin-1/2 XXZ chain initialized in a domain
wall state, where all spins to the left of the origin are up, all spins to its
right are down. The focus is on exact formulae, which hold for arbitrary finite
(real or imaginary) time. In particular, we compute the amplitudes
corresponding to the process where all but spins come back to their initial
orientation, as a fold contour integral. These results are obtained using a
correspondence with the six vertex model, and taking a somewhat complicated
Hamiltonian/Trotter-type limit. Several simple applications are studied and
also discussed in a broader context.Comment: 44 pages, 5 figure
Cyclic tridiagonal pairs, higher order Onsager algebras and orthogonal polynomials
The concept of cyclic tridiagonal pairs is introduced, and explicit examples
are given. For a fairly general class of cyclic tridiagonal pairs with
cyclicity N, we associate a pair of `divided polynomials'. The properties of
this pair generalize the ones of tridiagonal pairs of Racah type. The algebra
generated by the pair of divided polynomials is identified as a higher-order
generalization of the Onsager algebra. It can be viewed as a subalgebra of the
q-Onsager algebra for a proper specialization at q the primitive 2Nth root of
unity. Orthogonal polynomials beyond the Leonard duality are revisited in light
of this framework. In particular, certain second-order Dunkl shift operators
provide a realization of the divided polynomials at N=2 or q=i.Comment: 32 pages; v2: Appendices improved and extended, e.g. a proof of
irreducibility is added; v3: version for Linear Algebra and its Applications,
one assumption added in Appendix about eq. (A.2
A family of Nikishin systems with periodic recurrence coefficients
Suppose we have a Nikishin system of measures with the th generating
measure of the Nikishin system supported on an interval \Delta_k\subset\er
with for all . It is well known that
the corresponding staircase sequence of multiple orthogonal polynomials
satisfies a -term recurrence relation whose recurrence coefficients,
under appropriate assumptions on the generating measures, have periodic limits
of period . (The limit values depend only on the positions of the intervals
.) Taking these periodic limit values as the coefficients of a new
-term recurrence relation, we construct a canonical sequence of monic
polynomials , the so-called \emph{Chebyshev-Nikishin
polynomials}. We show that the polynomials themselves form a sequence
of multiple orthogonal polynomials with respect to some Nikishin system of
measures, with the th generating measure being absolutely continuous on
. In this way we generalize a result of the third author and Rocha
\cite{LopRoc} for the case . The proof uses the connection with block
Toeplitz matrices, and with a certain Riemann surface of genus zero. We also
obtain strong asymptotics and an exact Widom-type formula for the second kind
functions of the Nikishin system for .Comment: 30 pages, minor change