Statistics Of Knots And Entangled Random Walks
World Scientific Publishing Co Pte Ltd (Verlag)
978-981-02-2519-3 (ISBN)
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In this book, the author announces the class of problems called “entropy of knots” and gives an overview of modern physical applications of existing topological invariants.He constructs statistical models on knot diagrams and braids using the representations of Jones-Kauffman and Alexander invariants and puts forward the question of limit distribution of these invariants for randomly generated knots. The relation of powers of corresponding algebraic invariants to the Lyapunov exponents of the products of noncommutative matrices is described. Also the problem of conditional joint limit distributions for “brownian bridges” on braids is discussed. Special cases of noncommutative groups PSL(2,R), PSL(2,Z) and braid groups are considered in detail.In this volume, the author also discusses the application of conformal methods for explicit construction of topological invariants for random walks on multiconnected manifolds. The construction of these topological invariants and the monodromy properties of correlation function of some conformal theories are also discussed.The author also considers the physical applications of “knot entropy” problem in various physical systems, focussing on polymers.
Part 1 Knot diagrams as disordered spin systems: statistical problems in topology; review of abelian problems in statistics of entangled random walks and incompleteness of Gauss invariant; nonabelian knot invariants; lattice knot diagrams as disordered Potts model; annealed and quenched realizations of topoligical disorder. Part 2 Random walks on noncommutative groups: limit theorem for conditional distribution of products of independent unimodular 2X2 matrices; brownian bridges on braid groups and knot statistics; limit distributions of random loops on noncommutative groups. Part 3 Conformal methods in statistics of entangled random walks: construction of nonabelian connections for free and modular groups from conformal methods; monodromy properties of random walk on double pictured plane and relation to CFT; critical exponents for random walks in regular lattice of obstacles - random walk on Lobachevskii plane. Part 4 Critical properties of knots and entangled random walks and their physical applications: random walks in random array of topological obstacles; high elasticity of polymer networks; fractal structure of collapsed polymer loops in solutions and gels - "crumpled globule" formation; nematic phase transitions in entangled directed polymers.
Erscheint lt. Verlag | 1.9.1996 |
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Verlagsort | Singapore |
Sprache | englisch |
Themenwelt | Mathematik / Informatik ► Mathematik ► Statistik |
Naturwissenschaften ► Physik / Astronomie | |
ISBN-10 | 981-02-2519-9 / 9810225199 |
ISBN-13 | 978-981-02-2519-3 / 9789810225193 |
Zustand | Neuware |
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