2,916 research outputs found
Universal Lyndon Words
A word over an alphabet is a Lyndon word if there exists an
order defined on for which is lexicographically smaller than all
of its conjugates (other than itself). We introduce and study \emph{universal
Lyndon words}, which are words over an -letter alphabet that have length
and such that all the conjugates are Lyndon words. We show that universal
Lyndon words exist for every and exhibit combinatorial and structural
properties of these words. We then define particular prefix codes, which we
call Hamiltonian lex-codes, and show that every Hamiltonian lex-code is in
bijection with the set of the shortest unrepeated prefixes of the conjugates of
a universal Lyndon word. This allows us to give an algorithm for constructing
all the universal Lyndon words.Comment: To appear in the proceedings of MFCS 201
An explicit universal cycle for the (n-1)-permutations of an n-set
We show how to construct an explicit Hamilton cycle in the directed Cayley
graph Cay({\sigma_n, sigma_{n-1}} : \mathbb{S}_n), where \sigma_k = (1 2 >...
k). The existence of such cycles was shown by Jackson (Discrete Mathematics,
149 (1996) 123-129) but the proof only shows that a certain directed graph is
Eulerian, and Knuth (Volume 4 Fascicle 2, Generating All Tuples and
Permutations (2005)) asks for an explicit construction. We show that a simple
recursion describes our Hamilton cycle and that the cycle can be generated by
an iterative algorithm that uses O(n) space. Moreover, the algorithm produces
each successive edge of the cycle in constant time; such algorithms are said to
be loopless
Periodic-Orbit Theory of Universality in Quantum Chaos
We argue semiclassically, on the basis of Gutzwiller's periodic-orbit theory,
that full classical chaos is paralleled by quantum energy spectra with
universal spectral statistics, in agreement with random-matrix theory. For
dynamics from all three Wigner-Dyson symmetry classes, we calculate the
small-time spectral form factor as power series in the time .
Each term of that series is provided by specific families of pairs of
periodic orbits. The contributing pairs are classified in terms of close
self-encounters in phase space. The frequency of occurrence of self-encounters
is calculated by invoking ergodicity. Combinatorial rules for building pairs
involve non-trivial properties of permutations. We show our series to be
equivalent to perturbative implementations of the non-linear sigma models for
the Wigner-Dyson ensembles of random matrices and for disordered systems; our
families of orbit pairs are one-to-one with Feynman diagrams known from the
sigma model.Comment: 31 pages, 17 figure
Random graph states, maximal flow and Fuss-Catalan distributions
For any graph consisting of vertices and edges we construct an
ensemble of random pure quantum states which describe a system composed of
subsystems. Each edge of the graph represents a bi-partite, maximally entangled
state. Each vertex represents a random unitary matrix generated according to
the Haar measure, which describes the coupling between subsystems. Dividing all
subsystems into two parts, one may study entanglement with respect to this
partition. A general technique to derive an expression for the average
entanglement entropy of random pure states associated to a given graph is
presented. Our technique relies on Weingarten calculus and flow problems. We
analyze statistical properties of spectra of such random density matrices and
show for which cases they are described by the free Poissonian
(Marchenko-Pastur) distribution. We derive a discrete family of generalized,
Fuss-Catalan distributions and explicitly construct graphs which lead to
ensembles of random states characterized by these novel distributions of
eigenvalues.Comment: 37 pages, 24 figure
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