400 research outputs found
Arithmetic area for m planar Brownian paths
We pursue the analysis made in [1] on the arithmetic area enclosed by m
closed Brownian paths. We pay a particular attention to the random variable
S{n1,n2, ...,n} (m) which is the arithmetic area of the set of points, also
called winding sectors, enclosed n1 times by path 1, n2 times by path 2, ...,nm
times by path m. Various results are obtained in the asymptotic limit
m->infinity. A key observation is that, since the paths are independent, one
can use in the m paths case the SLE information, valid in the 1-path case, on
the 0-winding sectors arithmetic area.Comment: 12 pages, 2 figure
Statistics of reduced words in locally free and braid groups: Abstract studies and application to ballistic growth model
We study numerically and analytically the average length of reduced
(primitive) words in so-called locally free and braid groups. We consider the
situations when the letters in the initial words are drawn either without or
with correlations. In the latter case we show that the average length of the
reduced word can be increased or lowered depending on the type of correlation.
The ideas developed are used for analytical computation of the average number
of peaks of the surface appearing in some specific ballistic growth modelComment: 29 pages, LaTeX, 7 separated Postscript figures (available on
request), submitted to J. Phys. (A): Math. Ge
Localization effects in a periodic quantum graph with magnetic field and spin-orbit interaction
A general technique for the study of embedded quantum graphs with magnetic
fields and spin-orbit interaction is presented. The analysis is used to
understand the contribution of Rashba constant to the extreme localization
induced by magnetic field in the T3 shaped quantum graph. We show that this
effect is destroyed at generic values of the Rashba constant. On the other
hand, for certain combinations of the Rashba constant and the magnetic
parameters another series of infinitely degenerate eigenvalues appears.Comment: 25 pages, typos corrected, references extende
Algebraic and arithmetic area for planar Brownian paths
The leading and next to leading terms of the average arithmetic area enclosed by independent closed Brownian planar paths, with
a given length and starting from and ending at the same point, is
calculated. The leading term is found to be
and the -winding sector arithmetic area inside the paths is subleading
in the asymptotic regime. A closed form expression for the algebraic area
distribution is also obtained and discussed.Comment: 8 pages, 2 figure
Statistical Interparticle Potential of an Ideal Gas of Non-Abelian Anyons
We determine and study the statistical interparticle potential of an ideal
system of non-Abelian Chern-Simons (NACS) particles, comparing our results with
the corresponding results of an ideal gas of Abelian anyons. In the Abelian
case, the statistical potential depends on the statistical parameter and it has
a "quasi-bosonic" behaviour for statistical parameter in the range (0,1/2)
(non-monotonic with a minimum) and a "quasi-fermionic" behaviour for
statistical parameter in the range (1/2,1) (monotonically decreasing without a
minimum). In the non-Abelian case the behavior of the statistical potential
depends on the Chern- Simons coupling and the isospin quantum number: as a
function of these two parameters, a phase diagram with quasi-bosonic,
quasi-fermionic and bosonic-like regions is obtained and investigated. Finally,
using the obtained expression for the statistical potential, we compute the
second virial coefficient of the NACS gas, which correctly reproduces the
results available in literature.Comment: 21 pages, 4 color figure
On the spectrum of the Laplace operator of metric graphs attached at a vertex -- Spectral determinant approach
We consider a metric graph made of two graphs
and attached at one point. We derive a formula relating the
spectral determinant of the Laplace operator
in terms of the spectral
determinants of the two subgraphs. The result is generalized to describe the
attachment of graphs. The formulae are also valid for the spectral
determinant of the Schr\"odinger operator .Comment: LaTeX, 8 pages, 7 eps figures, v2: new appendix, v3: discussions and
ref adde
Exit and Occupation times for Brownian Motion on Graphs with General Drift and Diffusion Constant
We consider a particle diffusing along the links of a general graph
possessing some absorbing vertices. The particle, with a spatially-dependent
diffusion constant D(x) is subjected to a drift U(x) that is defined in every
point of each link. We establish the boundary conditions to be used at the
vertices and we derive general expressions for the average time spent on a part
of the graph before absorption and, also, for the Laplace transform of the
joint law of the occupation times. Exit times distributions and splitting
probabilities are also studied and several examples are discussed.Comment: Accepted for publication in J. Phys.
Windings of the 2D free Rouse chain
We study long time dynamical properties of a chain of harmonically bound
Brownian particles. This chain is allowed to wander everywhere in the plane. We
show that the scaling variables for the occupation times T_j, areas A_j and
winding angles \theta_j (j=1,...,n labels the particles) take the same general
form as in the usual Brownian motion. We also compute the asymptotic joint laws
P({T_j}), P({A_j}), P({\theta_j}) and discuss the correlations occuring in
those distributions.Comment: Latex, 17 pages, submitted to J. Phys.
Random Operator Approach for Word Enumeration in Braid Groups
We investigate analytically the problem of enumeration of nonequivalent
primitive words in the braid group B_n for n >> 1 by analysing the random word
statistics and the target space on the basis of the locally free group
approximation. We develop a "symbolic dynamics" method for exact word
enumeration in locally free groups and bring arguments in support of the
conjecture that the number of very long primitive words in the braid group is
not sensitive to the precise local commutation relations. We consider the
connection of these problems with the conventional random operator theory,
localization phenomena and statistics of systems with quenched disorder. Also
we discuss the relation of the particular problems of random operator theory to
the theory of modular functionsComment: 36 pages, LaTeX, 4 separated Postscript figures, submitted to Nucl.
Phys. B [PM
Area distribution of two-dimensional random walks on a square lattice
The algebraic area probability distribution of closed planar random walks of
length N on a square lattice is considered. The generating function for the
distribution satisfies a recurrence relation in which the combinatorics is
encoded. A particular case generalizes the q-binomial theorem to the case of
three addends. The distribution fits the L\'evy probability distribution for
Brownian curves with its first-order 1/N correction quite well, even for N
rather small.Comment: 8 pages, LaTeX 2e. Reformulated in terms of q-commutator
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