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Author Bovier, Anton
Title Mathematical Statistical Physics : Lecture Notes of the les Houches Summer School 2005
Imprint Oxford : Elsevier Science & Technology, 2006
©2006
book jacket
Edition 1st ed
Descript 1 online resource (849 pages)
text txt rdacontent
computer c rdamedia
online resource cr rdacarrier
Series Issn Ser. ; v.Volume 83
Issn Ser
Note Front cover -- Lecturers who contributed to this volume -- Title page -- Copyright page -- Previous sessions -- Organizers -- Lecturers -- Participants -- Preface -- Informal seminars -- Table of contents -- Course 1 Random matrices and determinantal processes -- Introduction -- Point processes -- General theory -- Determinantal processes -- Measures defined by products of several determinants -- Non-intersecting paths and the Aztec diamond -- Non-intersecting paths and the LGV theorem -- The Aztec diamond -- Relations to other models -- Asymptotics -- Double contour integral formula for the correlation kernel -- Asymptotics for the Aztec diamond -- Asymptotics for random permutations -- The corner growth model -- Mapping to non-intersecting paths -- The Schur and Plancherel measures -- A discrete polynuclear growth model -- Proof of theorem 5.1 -- References -- Course 2 Some recent aspects of random conformally invariant systems -- Some discrete models -- Self-avoiding walks and polygons -- Random walk loops -- Site-percolation -- The Ising model -- The Potts models -- FK representations of Potts models -- The O(N) models -- Conformal invariance -- A "conformal Haar measure" on self-avoiding loops -- Preliminaries -- A conformal invariance property -- Uniqueness -- Existence -- Schramm-Loewner Evolutions -- Definition -- Computing with SLE -- Conformal loop-ensembles -- Definition -- First properties -- The loop-soup construction -- The Gaussian free field -- Definition -- Cliffs" as level lines -- References -- Course 3 Conformal random geometry -- Preamble -- Introduction -- A brief conformal history -- Conformal geometrical structures -- Quantum gravity -- Stochastic Löwner evolution -- Recent developments -- Synopsis -- Intersections of random walks -- Non-intersection probabilities -- Quantum gravity -- Random walks on a random lattice
Non-intersections of packets of walks -- Mixing random & self-avoiding walks -- General star configurations -- Quantum gravity for SAW's & RW's -- RW-SAW exponents -- Brownian hiding exponents -- Percolation clusters -- Cluster hull and external perimeter -- Harmonic measure of percolation frontiers -- Harmonic and path crossing exponents -- Quantum gravity for percolation -- Multifractality of percolation clusters -- Conformally invariant frontiers and quantum gravity -- Harmonic measure and potential near a fractal frontier -- Calculation of multifractal exponents from quantum gravity -- Geometrical analysis of multifractal spectra -- Higher multifractal spectra -- Double-sided spectra -- Higher multifractality of multiple path vertices -- Winding of conformally invariant curves -- Harmonic measure and rotations -- Exact mixed multifractal spectra -- Conformal invariance and quantum gravity -- Rotation scaling exponents -- Legendre transform -- O(N) & Potts models and the Stochastic Löwner Evolution -- Geometric duality in O(N) and Potts cluster frontiers -- Geometric duality of SLEkappa -- Quantum gravity duality and SLE -- Dual dimensions -- KPZ for SLE -- Short distance expansion -- Multiple paths in O(N), Potts models and SLE -- SLE(kappa, rho) and quantum gravity -- Multifractal exponents for multiple SLE's -- References -- Course 4 Random motions in random media -- Introduction -- RWRE -- RCM -- Back to RWRE -- Diffusions in random environment -- References -- Course 5 An introduction to mean field spin glas theory: methods and results -- Introduction -- The mean field ferromagnetic model. Convexity and cavity methods -- The mean field spin glass model. Basic definitions -- The interpolation method and its generalizations -- The thermodynamic limit and the variational bounds -- The Parisi representation for the free energy
Conclusion and outlook for future developments -- References -- Course 6 Short-range spin glasses: selected open problems -- Introduction -- The Fortuin-Kasteleyn random cluster representation and phase transitions -- Spin glass ground states and invasion percolation -- Ground state multiplicity in the 2D EA spin glass -- References -- Course 7 Computing the number of metastable states in infinite-range models -- Introduction -- The TAP equations -- A simple analysis of the solutions of the Bethe equations -- The direct approach: general considerations -- The supersymmetric formulation -- Spontaneous supersymmetry breaking -- An explicit computation: the complexity of the SK model -- A few words on quenched complexity -- Conclusions and open problems -- References -- Course 8 Dynamics of trap models -- Introduction -- Definition of the Bouchaud trap model -- Examples of trap models -- Natural questions on trap models -- References -- The one-dimensional trap model -- The Fontes-Isopi-Newman singular diffusion -- The scaling limit -- Time-scale change of Brownian motion -- Convergence of the fixed-time distributions -- A coupling for walks on different scales -- Scaling limit -- Aging results -- Subaging results -- Behaviour of the aging functions on different time scales -- References -- The trap model in dimension larger than one -- The fractional-kinetics process -- Scaling limit -- Aging results -- The coarse-graining procedure -- References -- The arcsine law as a universal aging scheme -- Aging on large complete graphs -- Deep traps -- Shallow traps -- Very deep traps -- Proof of Theorem 5.1 -- The alpha-stable subordinator as a universal clock -- Potential-theoretic characterisation -- Applications of the arcsine law -- Aging in the REM -- Short time scales -- Long time scales -- Open questions and conjectures -- Aging on large tori
Appendix A. Subordinators -- References -- Course 9 Quantum entropy and quantum information -- Introduction -- Rudiments of Classical Information Theory -- Entropy in Classical Information Theory -- Entropy of a pair of random variables -- Shannon's Noiseless Channel Coding Theorem -- Asymptotic Equipartition Property (AEP) -- Consequences of the AEP -- Information transmission and Channel Capacity -- Introduction to Quantum Information Theory -- Open systems -- Properties of the density matrix -- Reduced density matrix and partial trace -- Time evolution of open systems -- Generalized measurements -- Implementation of a generalized measurement -- Quantum entropy -- Properties of the von Neumann entropy S(rho) -- Data compression in Quantum Information Theory -- Schumacher's Theorem for memoryless quantum sources -- Quantum channels and additivity -- The noise in the channel -- Capacities of a quantum channel -- Classical capacity of a quantum channel -- A sufficient condition for additivity -- Multiplicativity of the maximal p-norm -- Bibliography -- Course 10 Two lectures on iterative coding and statistical mechanics -- Introduction -- Codes on graphs -- A simple-minded bound and belief propagation -- Characterizing the code performances -- Bounding the conditional entropy -- A parenthesis -- Density evolution a.k.a. distributional recursive equations -- The area theorem and some general questions -- Historical and bibliographical note -- References -- Course 11 Evolution in fluctuating populations -- Introduction -- Some classical coalescent theory -- Kingman's coalescent -- Variable population size -- Introducing structure -- Fluctuations matter -- Balancing selection -- A second neutral locus -- The problem -- The ancestral recombination graph and local trees -- Back to the main plot -- The diffusion approximation -- Extensions
Summary and notes of caution -- Spatial structure and the Malécot formula -- Branching process models -- The classical stepping stone model -- Duality -- Neutral evolution -- Random walks in random environments -- Models in continuous space -- Malécot's formula -- Spatial models -- Locally regulated populations -- Competing species -- Branching annihilating random walk -- A duality and a conjecture for Model II -- Conjectures for Model I -- Heteromyopia -- References -- Course 12 Multi-scale analysis of population models -- Spatial diffusion models of population genetics -- Duality and representation via coalescent processes -- The function-valued dual process -- The refined dual process -- Application: ergodic theorem -- Historical processes -- The neutral models: IMM and IFWD -- The concept of the historical process -- The concept of the look-down process -- Representation theorem via look-down -- Representation via coalescent -- Quasi-equilibria: elements of the multi-scale analysis -- Mean-field models and McKean-Vlasov limit -- The mean-field dual -- Longtime behavior of McKean-Vlasov process -- Large time-space scale behavior of mean-field model -- Punctuated equilibria and hierarchical mean-field limit -- Detailed specification of selection and mutation mechanisms -- Interpretation of the model from the point of view of evolutionary theory -- The basic scenario -- The hierarchical mean-field limit -- Two interaction chains -- Basic limit theorem -- Punctuated equilibria: emergence and take-over -- References -- Course 13 Elements of nonequilibrium statistical mechanics -- Elements of introduction -- The goal -- The plan -- Elements of an H-theorem -- Heuristics of an H-theorem -- H-theorem: the problem -- Macroscopic H-theorem -- Semigroup property -- Autonomy on a finite partition -- Reversibility -- Propagation of constrained equilibrium
Pathwise H-theorem
The proceedings of the 2005 les Houches summer school on Mathematical Statistical Physics give and broad and clear overview on this fast developing area of interest to both physicists and mathematicians. · introduction to a field of math with many interdisciplinary connections in physics, biology, and computer science · roadmap to the next decade of mathematical statistical mechanics · volume for reference years to come
Description based on publisher supplied metadata and other sources
Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2020. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries
Link Print version: Bovier, Anton Mathematical Statistical Physics : Lecture Notes of the les Houches Summer School 2005 Oxford : Elsevier Science & Technology,c2006 9780444528131
Subject Mathematical physics -- Congresses.;Statistical mechanics -- Congresses
Electronic books
Alt Author Dunlop, François
Van Enter, Aernout
Den Hollander, Frank
Dalibard, Jean
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