21 research outputs found
Combinatorial interpretation and positivity of Kerov's character polynomials
Kerov's polynomials give irreducible character values in term of the free
cumulants of the associated Young diagram. We prove in this article a
positivity result on their coefficients, which extends a conjecture of S.
Kerov. Our method, through decomposition of maps, gives a description of the
coefficients of the k-th Kerov's polynomials using permutations in S(k). We
also obtain explicit formulas or combinatorial interpretations for some
coefficients. In particular, we are able to compute the subdominant term for
character values on any fixed permutation (it was known for cycles).Comment: 33 pages, 13 figures, version 3: minor modifcation
Free Meixner states
Free Meixner states are a class of functionals on non-commutative polynomials
introduced in math.CO/0410482. They are characterized by a resolvent-type form
for the generating function of their orthogonal polynomials, by a recursion
relation for those polynomials, or by a second-order non-commutative
differential equation satisfied by their free cumulant functional. In this
paper, we construct an operator model for free Meixner states. By combinatorial
methods, we also derive an operator model for their free cumulant functionals.
This, in turn, allows us to construct a number of examples. Many of these
examples are shown to be trivial, in the sense of being free products of
functionals which depend on only a single variable, or rotations of such free
products. On the other hand, the multinomial distribution is a free Meixner
state and is not a product. Neither is a large class of tracial free Meixner
states which are analogous to the simple quadratic exponential families in
statistics.Comment: 30 page
On the large N limit of matrix integrals over the orthogonal group
We reexamine the large N limit of matrix integrals over the orthogonal group
O(N) and their relation with those pertaining to the unitary group U(N). We
prove that lim_{N to infty} N^{-2} \int DO exp N tr JO is half the
corresponding function in U(N), and a similar relation for lim_{N to infty}
\int DO exp N tr(A O B O^t), for A and B both symmetric or both skew symmetric.Comment: 12 page
Meixner class of non-commutative generalized stochastic processes with freely independent values I. A characterization
Let be an underlying space with a non-atomic measure on it (e.g.
and is the Lebesgue measure). We introduce and study a
class of non-commutative generalized stochastic processes, indexed by points of
, with freely independent values. Such a process (field),
, , is given a rigorous meaning through smearing out
with test functions on , with being a
(bounded) linear operator in a full Fock space. We define a set
of all continuous polynomials of , and then define a con-commutative
-space by taking the closure of in the norm
, where is the vacuum in the Fock
space. Through procedure of orthogonalization of polynomials, we construct a
unitary isomorphism between and a (Fock-space-type) Hilbert space
, with
explicitly given measures . We identify the Meixner class as those
processes for which the procedure of orthogonalization leaves the set invariant. (Note that, in the general case, the projection of a
continuous monomial of oder onto the -th chaos need not remain a
continuous polynomial.) Each element of the Meixner class is characterized by
two continuous functions and on , such that, in the
space, has representation
\omega(t)=\di_t^\dag+\lambda(t)\di_t^\dag\di_t+\di_t+\eta(t)\di_t^\dag\di^2_t,
where \di_t^\dag and \di_t are the usual creation and annihilation
operators at point
Circular Law Theorem for Random Markov Matrices
Consider an nxn random matrix X with i.i.d. nonnegative entries with bounded
density, mean m, and finite positive variance sigma^2. Let M be the nxn random
Markov matrix with i.i.d. rows obtained from X by dividing each row of X by its
sum. In particular, when X11 follows an exponential law, then M belongs to the
Dirichlet Markov Ensemble of random stochastic matrices. Our main result states
that with probability one, the counting probability measure of the complex
spectrum of n^(1/2)M converges weakly as n tends to infinity to the uniform law
on the centered disk of radius sigma/m. The bounded density assumption is
purely technical and comes from the way we control the operator norm of the
resolvent.Comment: technical update via http://HAL.archives-ouvertes.f
Dynamic response of a micro-periodic beam under moving load-deterministic and stochastic approach
In the paper, the deterministic and stochastic approach to the problem of vibrations of a beam with periodically varying geometry under moving load is presented. A new averaged model for the dynamics of the periodic-like beam with a variable cross-section, Mazur-Śniady (2001), is applied. The approach to dynamics of the periodic-like beam assumed in the paper is based on concepts of the tolerance-averaged model by Woźniak (1999). The solution obtained for a single moving force is the basis of solution of stochastic vibrations caused by random train of moving forces.Deterministyczne i stochastyczne drgania belki o okresowo zmiennej geometrii wywołane ruchomym obciążeniem. W pracy rozpatruje się drgania belki o okresowo zmiennej geometrii wywołane działaniem ruchomych obciążeń. Wykorzystuje się model belko o prawie peridyczne strukturze (Mazur- Śniady, 2001), otrzymany metodą uśredniania tolerancyjnego (Woźniak, 1999). Podano rozwiązanie zagadnienia drgań belki o okresowo zmiennej sztywności wywołanych poruszającą się ze stałą prędkością siłą skupioną. Powyższe rozwiązanie wykorzystano wyznaczając probabilistyczne charakterystyki przemieszczeń belki obciążonej losowym ciągiem ruchomych sił skupionych