From random walks to spin glasses

Abstract The talk was a short review on systems which exhibit non-self-averaging effects: sums of random variables when the distribution has a long tail, mean field spin glasses, random map models and returns of a random walk to the origin. Non-self-averaging effects are identical in the case of sums of random variables and in the spin glass problem as predicted by the replica approach. Also we will see that for the random map models or for the problem of the returns of a random walk to the origin, the non-self-averaging effects coincide with the results of the replica approach when the number n of replica n = − 1 2 or n = −1.

[1]  Bernard Derrida,et al.  Statistical properties of randomly broken objects and of multivalley structures in disordered systems , 1987 .

[2]  Giorgio Parisi,et al.  Order parameter for spin-glasses , 1983 .

[3]  D. Sherrington.  Ising replica magnets , 1980 .

[4]  Bernard Derrida,et al.  The random map model: a disordered model with deterministic dynamics , 1987 .

[5]  M. Mézard,et al.  Replica symmetry breaking and the nature of the spin glass phase , 1984 .

[6]  B. Derrida.  Random-energy model: An exactly solvable model of disordered systems , 1981 .

[7]  R. W. Penney,et al.  Coupled dynamics of fast spins and slow interactions in neural networks and spin systems , 1993 .

[8]  A. Blandin.  THEORIES VERSUS EXPERIMENTS IN THE SPIN GLASS SYSTEMS , 1978 .

[9]  A. E. Patrick,et al.  Directed polymers on trees: a martingale approach , 1993 .

[10]  D. Thouless,et al.  Stability of the Sherrington-Kirkpatrick solution of a spin glass model , 1978 .

[11]  A. Vershik,et al.  Limit Measures Arising in the Asympyotic Theory of Symmetric Groups. I. , 1977 .

[12]  G. Parisi.  The order parameter for spin glasses: a function on the interval 0-1 , 1980 .

[13]  Bernard Derrida,et al.  The probability distribution of the partition function of the random energy model , 1989 .

[14]  Bernard Derrida,et al.  Multivalley structure in Kauffman's model: analogy with spin glasses , 1986 .

[15]  Higgs.  Frequency distributions in population genetics parallel those in statistical physics. , 1995, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[16]  J. Bouchaud,et al.  Anomalous diffusion in disordered media: Statistical mechanisms, models and physical applications , 1990 .

[17]  Sample to sample fluctuations in the random energy model , 1985 .

[18]  S. Kirkpatrick,et al.  Solvable Model of a Spin-Glass , 1975 .

[19]  Bernard Derrida,et al.  Statistical properties of valleys in the annealed random map model , 1988 .

[20]  D. Ruelle.  A mathematical reformulation of Derrida's REM and GREM , 1987 .

[21]  M. Mézard,et al.  Spin Glass Theory and Beyond , 1987 .

[22]  G. Parisi.  On the branching structure of the tree of states in spin glasses , 1993 .

[23]  S. Kirkpatrick,et al.  Infinite-ranged models of spin-glasses , 1978 .

[24]  M. A. Virasoro,et al.  Random free energies in spin glasses , 1985 .

[25]  J. Pitman,et al.  Random Discrete Distributions Derived from Self-Similar Random Sets , 1996 .

[26]  I. Kondor.  Parisi's mean-field solution for spin glasses as an analytic continuation in the replica number , 1983 .

[27]  B. Derrida.  DIRECTED POLYMERS IN A RANDOM MEDIUM , 1990 .

[28]  B. Derrida.  Non-Self-Averaging Effects in Sums of Random Variables, Spin Glasses, Random Maps and Random Walks , 1994 .

[29]  R. Palmer,et al.  The replica method and solvable spin glass model , 1979 .

[30]  Bernard Derrida,et al.  Magnetic properties and the function q(x) of the generalised random-energy model , 1986 .