Algebraic and arithmetic area for mm planar Brownian paths

The leading and next to leading terms of the average arithmetic area $< S(m)>$ enclosed by $m\to\infty$ independent closed Brownian planar paths, with a given length $t$ and starting from and ending at the same point, is calculated. The leading term is found to be $\sim {\pi t\over 2}\ln m$ and the $0$-winding sector arithmetic area inside the $m$ 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

Scattering theory on graphs

We consider the scattering theory for the Schr\"odinger operator -\Dc_x^2+V(x) on graphs made of one-dimensional wires connected to external leads. We derive two expressions for the scattering matrix on arbitrary graphs. One involves matrices that couple arcs (oriented bonds), the other involves matrices that couple vertices. We discuss a simple way to tune the coupling between the graph and the leads. The efficiency of the formalism is demonstrated on a few known examples.Comment: 21 pages, LaTeX, 10 eps figure

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

Geometric Exponents, SLE and Logarithmic Minimal Models

In statistical mechanics, observables are usually related to local degrees of freedom such as the Q < 4 distinct states of the Q-state Potts models or the heights of the restricted solid-on-solid models. In the continuum scaling limit, these models are described by rational conformal field theories, namely the minimal models M(p,p') for suitable p, p'. More generally, as in stochastic Loewner evolution (SLE_kappa), one can consider observables related to nonlocal degrees of freedom such as paths or boundaries of clusters. This leads to fractal dimensions or geometric exponents related to values of conformal dimensions not found among the finite sets of values allowed by the rational minimal models. Working in the context of a loop gas with loop fugacity beta = -2 cos(4 pi/kappa), we use Monte Carlo simulations to measure the fractal dimensions of various geometric objects such as paths and the generalizations of cluster mass, cluster hull, external perimeter and red bonds. Specializing to the case where the SLE parameter kappa = 4p'/p is rational with p < p', we argue that the geometric exponents are related to conformal dimensions found in the infinitely extended Kac tables of the logarithmic minimal models LM(p,p'). These theories describe lattice systems with nonlocal degrees of freedom. We present results for critical dense polymers LM(1,2), critical percolation LM(2,3), the logarithmic Ising model LM(3,4), the logarithmic tricritical Ising model LM(4,5) as well as LM(3,5). Our results are compared with rigourous results from SLE_kappa, with predictions from theoretical physics and with other numerical experiments. Throughout, we emphasize the relationships between SLE_kappa, geometric exponents and the conformal dimensions of the underlying CFTs.Comment: Added reference

Elementary derivation of Spitzer's asymptotic law for Brownian windings and some of its physical applications

A simple derivation of Spitzer'z asymptotic law for Brownian windings [Trans.Am.Math.Soc.87,187 (1958)]is presented along with its generalizations >.These include the cases of planar Brownian walks interacting with a single puncture and Brownian walks on a single truncated cone with variable conical angle interacting with the truncated conical tip.Such situations are typical in the theories of quantum Hall effect and 2+1 quantum gravity, respectively .They also have some applications in polymer physic

Finite pseudo orbit expansions for spectral quantities of quantum graphs

We investigate spectral quantities of quantum graphs by expanding them as sums over pseudo orbits, sets of periodic orbits. Only a finite collection of pseudo orbits which are irreducible and where the total number of bonds is less than or equal to the number of bonds of the graph appear, analogous to a cut off at half the Heisenberg time. The calculation simplifies previous approaches to pseudo orbit expansions on graphs. We formulate coefficients of the characteristic polynomial and derive a secular equation in terms of the irreducible pseudo orbits. From the secular equation, whose roots provide the graph spectrum, the zeta function is derived using the argument principle. The spectral zeta function enables quantities, such as the spectral determinant and vacuum energy, to be obtained directly as finite expansions over the set of short irreducible pseudo orbits.Comment: 23 pages, 4 figures, typos corrected, references added, vacuum energy calculation expande

Functionals of the Brownian motion, localization and metric graphs

We review several results related to the problem of a quantum particle in a random environment. In an introductory part, we recall how several functionals of the Brownian motion arise in the study of electronic transport in weakly disordered metals (weak localization). Two aspects of the physics of the one-dimensional strong localization are reviewed : some properties of the scattering by a random potential (time delay distribution) and a study of the spectrum of a random potential on a bounded domain (the extreme value statistics of the eigenvalues). Then we mention several results concerning the diffusion on graphs, and more generally the spectral properties of the Schr\"odinger operator on graphs. The interest of spectral determinants as generating functions characterizing the diffusion on graphs is illustrated. Finally, we consider a two-dimensional model of a charged particle coupled to the random magnetic field due to magnetic vortices. We recall the connection between spectral properties of this model and winding functionals of the planar Brownian motion.Comment: Review article. 50 pages, 21 eps figures. Version 2: section 5.5 and conclusion added. Several references adde

Studies in the statistical and thermal properties of hadronic matter under some extreme conditions

The thermal and statistical properties of hadronic matter under some extreme conditions are investigated using an exactly solvable canonical ensemble model. A unified model describing both the fragmentation of nuclei and the thermal properties of hadronic matter is developed. Simple expressions are obtained for quantities such as the hadronic equation of state, specific heat, compressibility, entropy, and excitation energy as a function of temperature and density. These expressions encompass the fermionic aspect of nucleons, such as degeneracy pressure and Fermi energy at low temperatures and the ideal gas laws at high temperatures and low density. Expressions are developed which connect these two extremes with behavior that resembles an ideal Bose gas with its associated Bose condensation. In the thermodynamic limit, an infinite cluster exists below a certain critical condition in a manner similar to the sudden appearance of the infinite cluster in percolation theory. The importance of multiplicity fluctuations is discussed and some recent data from the EOS collaboration on critical point behavior of nuclei can be accounted for using simple expressions obtained from the model.Comment: 22 pages, revtex, includes 6 figures, submitted to Phys. Rev.