We study a certain random groeth model in two dimensions closely related to the one-dimensional totally asymmetric exclusion process. The results show that the shape fluctuations, appropriately scaled, converges in distribution to the Tracy-Widom largest eigenvalue distribution for the Gaussian Unitary Ensemble.

Shape Fluctuations and Random Matrices

Recent works have shown that the family of probability distributions with moments given by the Fuss–Catalan numbers permit a simple parameterized form for their density. We extend this result to the Raney distribution which by definition has its moments given by a generalization of the Fuss–Catalan numbers. Such computations begin with an algebraic equation satisfied by the Stieltjes transform, which we show can be derived from the linear differential equation satisfied by the characteristic polynomial of random matrix realizations of the Raney distribution. For the Fuss–Catalan distribution, an equilibrium problem characterizing the density is identified. The Stieltjes transform for the limiting spectral density of the singular values squared of the matrix product formed from $q$ inverse standard Gaussian matrices, and $s$ standard Gaussian matrices, is shown to satisfy a variant of the algebraic equation relating to the Raney distribution. Supported on $[0,\infty )$, we show that it too permits a simple functional form upon the introduction of an appropriate choice of parameterization. As an application, the leading asymptotic form of the density as the endpoints of the support are approached is computed, and is shown to have some universal features.

/pdf/raney-distributions-and-random-matrix-theory-4z64h1mfpq.pdf

Raney Distributions and Random Matrix Theory

The study of the atmospheric pollen content is very interesting for the allergist, allowing it to specify the nature and the quality of the pollens that contained in the atmosphere, this according to the seasons. This atmospheric content is important both quantitatively because of the allergenic pollen must be widespread, and qualitatively because of the pollen that causes respiratory allergies which belongs to a few well-known families (Grasses (Poaceae), Composeae, Chenopodiaceae, Urticaceae, Platanaceae, Plantaginaceae and Polygonaceae (Touraine et al., 1969). Our work focuses on the pollen identification in the Guelma region, in order to establish a pollen calendar, and to determine which pollens can cause allergies in sensitive subjects and to understand the influence of pollen meteorological conditions on atmospheric pollen concentrations. Our study is a part of an area where palynological studies are non-existent. It, therefore, provided the first data on the atmospheric pollen content of the Guelma region (North-Eastern Algeria). The gravimetric method used in the collection of airborne pollen grains which allowed us to capture a total of 3348 pollen grains (GP) and to identify 66 taxa representing 33 plant families. The most represented families are Cupressaceae with 755 GP (22.55%), Poaceae in second place (649 GP or 19.38%) and third-order Pinaceae (434 GP, or 13,08%). Among the 33 families are surveyed, 18 families were known for their allergenicity, ie 65% of the total number of families surveyed. Some of them have a high allergenicity (Cupressaceae, Poaceae, Oleaceae, and Betulaceae). A total of 2186 GPs were counted with anemophilous pollination, ie 65% of the total quantity recorded, 756 entomophilious GP (23%) and 406 GP with mixed pollination (anemophilous and entomophilious) or 12% of the total harvest. The data which obtained from the identification and counting of the pollen grains during the eight months of study, it has been used to realize the pollen calendar of our studied region. This study revealed the effect of meteorological conditions on the atmospheric pollen concentrations: temperature, precipitation, air humidity and wind speed. Finally, this first approach will have to be supplemented and deepened by the use of new capture methods (volumetric methods); similarly, reference collections will be needed for the identification of the atmospheric pollens.

Republique algerienne democratique et populaire ministere de l'enseignement superieur et de la recherche scientifique universite aboubekr belkaid tlemcen faculte des sciences de la nature et de la vie et sciences de la terre et de l'univers

An invariant ensemble of $N\times N$ random matrices can be characterised by a joint distribution for eigenvalues $P(\lambda_1,\cdots,\lambda_N)$. The study of the distribution of linear statistics, i.e. of quantities of the form $L=(1/N)\sum_if(\lambda_i)$ where $f(x)$ is a given function, appears in many physical problems. In the $N\to\infty$ limit, $L$ scales as $L\sim N^\eta$, where the scaling exponent $\eta$ depends on the ensemble and the function $f$. Its distribution can be written under the form $P_N(s=N^{-\eta}\,L)\simeq A_{\beta,N}(s)\,\exp\big\{-(\beta N^2/2)\,\Phi(s)\big\}$, where $\beta\in\{1,\,2,\,4\}$ is the Dyson index. The Coulomb gas technique naturally provides the large deviation function $\Phi(s)$, which can be efficiently obtained thanks to a "thermodynamic identity" introduced earlier. We conjecture the pre-exponential function $A_{\beta,N}(s)$. We check our conjecture on several well controlled cases within the Laguerre and the Jacobi ensembles. Then we apply our main result to a situation where the large deviation function has no minimum (and $L$ has infinite moments)~: this arises in the statistical analysis of the Wigner time delay for semi-infinite multichannel disordered wires (Laguerre ensemble). The statistical analysis of the Wigner time delay then crucially depends on the pre-exponential function $A_{\beta,N}(s)$, which ensures the decay of the distribution for large argument.

Distribution of spectral linear statistics on random matrices beyond the large deviation function—Wigner time delay in multichannel disordered wires

The eigenvalues of the matrix structure X + X(0), where X is a random Gaussian Hermitian matrix and X(0) is non-random or random independent of X, are closely related to Dyson Brownian motion. Previous works have shown how an infinite hierarchy of equations satisfied by the dynamical correlations become triangular in the infinite density limit, and give rise to the complex Burgers equation for the Greens function of the corresponding one-point density function. We show how this and analogous partial differential equations, for chiral, circular and Jacobi versions of Dyson Brownian motion follow from a macroscopic hydrodynamical description involving the current density and continuity equation. The method of characteristics gives a systematic approach to solving the PDEs, and in the chiral case we show how this efficiently reclaims the characterization of the global eigenvalue density for non-central Wishart matrices due to Dozier and Silverstein. Collective variables provide another approach to deriving the complex Burgers equation in the Gaussian case, and we show that this approach applies equally as well to chiral matrices. We relate both the Gaussian and chiral cases to the asymptotics of matrix integrals.

Hydrodynamical spectral evolution for random matrices

Cette these est consacree a l'etude des matrices aleatoires et a quelques unes de leurs applications en physique, en particulier en physique statistique et en physique quantique.C'est un travail essentiellement analytique complete par quelques simulations numeriques Monte Carlo. Dans un premier temps j'introduis la theorie des matrices aleatoires de facon assez generale : je definis les principaux ensembles de matrices aleatoires (en particulier gaussiens) et decris leurs proprietes fondamentales (distribution des valeurs propres, densite, etc). Dans un second temps je m'interesse a des systemes physiques d'interfaces a l'equilibre qui peuvent etre modelises par des marcheurs ``vicieux'', c'est-a-dire des marcheurs aleatoires conditionnes a ne pas se croiser. On peut montrer que la distribution des positions des marcheurs a un temps donne est exactement celle des valeurs propres d'une matrice aleatoire. J'etudie ensuite un probleme physique qui releve d'un domaine tres different, celui de l'information quantique, mais qui est egalement etroitement relie aux matrices aleatoires: celui de l'intrication pour des etats aleatoires dans un systeme quantique bipartite (fait de deux sous-parties) de grande taille. Enfin je m'interesse a certaines proprietes des matrices aleatoires comme la distribution du nombre de valeurs propres positives ou encore la distribution de la valeur propre maximale (loi de Tracy-Widom pres de la moyenne et grandes deviations loin de la moyenne).

https://tel.archives-ouvertes.fr/tel-00633266/document

Matrices aléatoires et leurs applications à la physique statistique et quantique

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Matrices aléatoires et leurs applications à la physique statistique et quantique