51 research outputs found
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
Symmetric function generalizations of graph polynomials
Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1995.Includes bibliographical references (p. 67-70).by Timothy Yi-Chung Chow.Ph.D
Random Matrix Theories in Quantum Physics: Common Concepts
We review the development of random-matrix theory (RMT) during the last
decade. We emphasize both the theoretical aspects, and the application of the
theory to a number of fields. These comprise chaotic and disordered systems,
the localization problem, many-body quantum systems, the Calogero-Sutherland
model, chiral symmetry breaking in QCD, and quantum gravity in two dimensions.
The review is preceded by a brief historical survey of the developments of RMT
and of localization theory since their inception. We emphasize the concepts
common to the above-mentioned fields as well as the great diversity of RMT. In
view of the universality of RMT, we suggest that the current development
signals the emergence of a new "statistical mechanics": Stochasticity and
general symmetry requirements lead to universal laws not based on dynamical
principles.Comment: 178 pages, Revtex, 45 figures, submitted to Physics Report
Geometric and Topological Combinatorics
The 2007 Oberwolfach meeting “Geometric and Topological Combinatorics” presented a great variety of investigations where topological and algebraic methods are brought into play to solve combinatorial and geometric problems, but also where geometric and combinatorial ideas are applied to topological questions
Alternating sign matrices and polytopes
This thesis deals with two types of mathematical objects: alternating sign matrices and polytopes. Alternating sign matrices were first defined in 1982 by Mills, Robbins and Rumsey. Since then, alternating sign matrices have led to some very captivating research (with multiple open problems still standing), an outline of which is presented in the opening chapter of this thesis. Convex polytopes are extremely relevant when considering enumerations of certain classes of integer valued matrices. An overview of the relevant properties of convex polytopes is presented, before a connection is made between polytopes and alternating sign matrices: the alternating sign matrix polytope. The vertex set of this new polytope is given, as well as a generalization of standard alternating sign matrices to give higher spin alternating sign matrices. From a result of Ehrhart a result concerning the enumeration of these matrices is obtained, namely, that for fixed size and variable line sum the enumeration is given by a particular polynomial. In Chapter 4, we give results concerning the symmetry classes of the alternating sign matrix polytope and in Chapter 3 we study symmetry classes of the Birkhoff polytope. For this classical polytope we give some new results. In the penultimate chapter, another polytope is defined that is a valid solution set of the transportation problem and for which a particular set of parameters gives the alternating sign matrix polytope. Importantly the transportation polytope is a subset of this new polytope.EThOS - Electronic Theses Online ServiceGBUnited Kingdo
Alternating sign matrices and polytopes
This thesis deals with two types of mathematical objects: alternating sign matrices and polytopes. Alternating sign matrices were first defined in 1982 by Mills, Robbins and Rumsey. Since then, alternating sign matrices have led to some very captivating research (with multiple open problems still standing), an outline of which is presented in the opening chapter of this thesis. Convex polytopes are extremely relevant when considering enumerations of certain classes of integer valued matrices. An overview of the relevant properties of convex polytopes is presented, before a connection is made between polytopes and alternating sign matrices: the alternating sign matrix polytope. The vertex set of this new polytope is given, as well as a generalization of standard alternating sign matrices to give higher spin alternating sign matrices. From a result of Ehrhart a result concerning the enumeration of these matrices is obtained, namely, that for fixed size and variable line sum the enumeration is given by a particular polynomial. In Chapter 4, we give results concerning the symmetry classes of the alternating sign matrix polytope and in Chapter 3 we study symmetry classes of the Birkhoff polytope. For this classical polytope we give some new results. In the penultimate chapter, another polytope is defined that is a valid solution set of the transportation problem and for which a particular set of parameters gives the alternating sign matrix polytope. Importantly the transportation polytope is a subset of this new polytope.EThOS - Electronic Theses Online ServiceGBUnited Kingdo
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