Phase transitions in diluted negative-weight percolation models

We investigate the geometric properties of loops on two-dimensional lattice graphs, where edge weights are drawn from a distribution that allows for positive and negative weights. We are interested in the appearance of spanning loops of total negative weight. The resulting percolation problem is fundamentally different from conventional percolation, as we have seen in a previous study of this model for the undiluted case. Here, we investigate how the percolation transition is affected by additional dilution. We consider two types of dilution: either a certain fraction of edges exhibit zero weight, or a fraction of edges is even absent. We study these systems numerically using exact combinatorial optimization techniques based on suitable transformations of the graphs and applying matching algorithms. We perform a finite-size scaling analysis to obtain the phase diagram and determine the critical properties of the phase boundary. We find that the first type of dilution does not change the universality class compared to the undiluted case whereas the second type of dilution leads to a change of the universality class.Comment: 8 pages, 7 figure

D-particles, Matrix Integrals and KP hierachy

We study the regularized correlation functions of the light-like coordinate operators in the reduction to zero dimensions of the matrix model describing $D$-particles in four dimensions. We investigate in great detail the related matrix model originally proposed and solved in the planar limit by J. Hoppe. It also gives the solution of the problem of 3-coloring of planar graphs. We find interesting strong/weak 't Hooft coupling dependence. The partition function of the grand canonical ensemble turns out to be a tau-function of KP hierarchy. As an illustration of the method we present a new derivation of the large-N and double-scaling limits of the one-matrix model with cubic potential.Comment: harvmac, 35 pp. v2. typos correcte

Flipturning polygons

A flipturn is an operation that transforms a nonconvex simple polygon into another simple polygon, by rotating a concavity 180 degrees around the midpoint of its bounding convex hull edge. Joss and Shannon proved in 1973 that a sequence of flipturns eventually transforms any simple polygon into a convex polygon. This paper describes several new results about such flipturn sequences. We show that any orthogonal polygon is convexified after at most n-5 arbitrary flipturns, or at most 5(n-4)/6 well-chosen flipturns, improving the previously best upper bound of (n-1)!/2. We also show that any simple polygon can be convexified by at most n^2-4n+1 flipturns, generalizing earlier results of Ahn et al. These bounds depend critically on how degenerate cases are handled; we carefully explore several possibilities. We describe how to maintain both a simple polygon and its convex hull in O(log^4 n) time per flipturn, using a data structure of size O(n). We show that although flipturn sequences for the same polygon can have very different lengths, the shape and position of the final convex polygon is the same for all sequences and can be computed in O(n log n) time. Finally, we demonstrate that finding the longest convexifying flipturn sequence of a simple polygon is NP-hard.Comment: 26 pages, 32 figures, see also http://www.uiuc.edu/~jeffe/pubs/flipturn.htm

Pattern Theorem for the Hexagonal Lattice

Master'sMASTER OF SCIENC

Knotting statistics after a local strand passage in unknotted self-avoiding polygons in Z3

We study here a model for a strand passage in a ring polymer about a randomly chosen location at which two strands of the polymer have been brought gclose h together. The model is based on ƒ¦-SAPs, which are unknotted self-avoiding polygons in Z^3 that contain a fixed structure ƒ¦ that forces two segments of the polygon to be close together. To study this model, the Composite Markov Chain Monte Carlo (CMCMC) algorithm, referred to as the CMC ƒ¦-BFACF algorithm, that I developed and proved to be ergodic for unknotted ƒ¦-SAPs in my M. Sc. Thesis, is used. Ten simulations (each consisting of 9.6 ~10^10 time steps) of the CMC ƒ¦-BFACF algorithm are performed and the results from a statistical analysis of the simulated data are presented. To this end, a new maximum likelihood method, based on previous work of Berretti and Sokal, is developed for obtaining maximum likelihood estimates of the growth constants and critical exponents associated respectively with the numbers of unknotted (2n)-edge ƒ¦-SAPs, unknotted (2n)-edge successful-strand-passage ƒ¦-SAPs, unknotted (2n)-edge failed-strand-passage ƒ¦-SAPs, and (2n)-edge after-strand-passage-knot-type-K unknotted successful-strand-passage ƒ¦-SAPs. The maximum likelihood estimates are consistent with the result (proved here) that the growth constants are all equal, and provide evidence that the associated critical exponents are all equal. We then investigate the question gGiven that a successful local strand passage occurs at a random location in a (2n)-edge knot-type K ƒ¦-SAP, with what probability will the ƒ¦-SAP have knot-type K f after the strand passage? h. To this end, the CMCMC data is used to obtain estimates for the probability of knotting given a (2n)-edge successful-strand-passage ƒ¦-SAP and the probability of an after-strand-passage polygon having knot-type K given a (2n)-edge successful-strand-passage ƒ¦-SAP. The computed estimates numerically support the unproven conjecture that these probabilities, in the n ¨ ‡ limit, go to a value lying strictly between 0 and 1. We further prove here that the rate of approach to each of these limits (should the limits exist) is less than exponential. We conclude with a study of whether or not there is a difference in the gsize h of an unknotted successful-strand-passage ƒ¦-SAP whose after-strand-passage knot-type is K when compared to the gsize h of a ƒ¦-SAP whose knot-type does not change after strand passage. The two measures of gsize h used are the expected lengths of, and the expected mean-square radius of gyration of, subsets of ƒ¦-SAPs. How these two measures of gsize h behave as a function of a polygon fs length and its after-strand-passage knot-type is investigated

Desmoplakin and Plakophilin-1a: Structure, subcellular distribution, and interactions

Memoria presentada por Ana María Carballido Vázquez para optar al grado de doctor por la Universidad de Salamanca.Ana M Carballido Vázquez ha realizado esta tesis doctoral siendo beneficiaria de una ayuda predoctoral del programa de Formación de Personal Investigador (BES-2010- 038674) del Ministerio de Economía y Competitividad, durante el periodo de septiembre de 2010 a agosto de 2014. Este trabajo se ha enmarcado dentro de los proyectos del Plan nacional I+D+i “Bases estructurales de interacciones en hemidesmosomas: integrina α6β4, BPAG1e y tetraspanina CD151” (BFU2009-08389) y “Bases estructurales de la función de plakinas en adhesión celular, implicación en enfermedades” (BFU2012-32847) financiados por el Ministerio de Ciencia e Innovación, el Ministerio de Economía y Competitividad y el Fondo Europeo de Desarrollo Regional; y del proyecto del Programa de apoyo a proyectos de investigación de la Consejería de Educación de la junta de Castilla y León titulado “Plakinas en complejos de adhesión: bases estructurales de su función e identificación de compuestos moduladores” (CSI181A12-1). Ana M Carballido Vázquez realizó parte del trabajo de esta tesis durante una estancia de ocho meses en el laboratorio del Prof. Arnoud Sonnenberg en el Netherlands Cancer Institute (NKI) (Ámsterdam, Holanda). Parte de esa estancia estuvo financiada por una ayuda de Estancia Breve (referencia EEBB-I-13-06399) del Ministerio de Economía y Competitividad.Peer Reviewe

Bosonic loop soups and their occupation fields

We consider a model for random loops on graphs which is inspired by the Feynman–Kac formula for the grand canonical partition function of an ideal gas. We associate to this model a corresponding occupation field, which is a positive random field detailing the total time spent by loops at each vertex. We argue that well known critical phenomena for the ideal gas can be reinterpreted in terms of random variables of this occupation field. We also argue that higher order correlations, such as the existence of off-diagonal long-range order, can only be seen in the occupation field by studying a modified space–time model of loops. We provide an isomorphism theorem for this model to a complex Gaussian field, and derive a version of Symanzik’s formula which describes the ideal gas interacting with a random background environment. Finally we consider the effect of interactions on the gas, and present a large deviations analysis of the cycle distribution of the loop model under two mean field Hamiltonians

Self-Avoiding Random Loops

A random loop, or polygon, is a simple random walk whose trajectory is a simple, closed Jordan Curve. This article contributes to the study of random loops in two ways. First, it initiates the study of the probability P n (x; y) that a random n-step loop contains a point (x; y) in the interior of the loop and shows that p n ( 1 2 ; 1 2 ) is 1 2 \Gamma 1 n . It is plausible that P n (x; y) % 1 2 for all (x; y), but we do not know this even for (x; y) = ( 3 2 ; 1 2 ). Second, this note offers a way to simulate random n-step self-avoiding loops. Numerical evidence obtained with this simulation procedure suggests that P n ( 3 2 ; 1 2 ) ß 1 2 \Gamma c n . Here is where Steve Rice's particular magic would have been especially useful. Department of Mathematics, University of California, Berkeley. Research partially supported by NSF Grant DMS 86-01634 y AT&T Bell Laboratories, 600 Mountain Av., Murray Hill, NJ 07974 1 Introduction Walks on the 2-dimensional rectangu..