ISRN Probability and Statistics

Volume 2013 (2013), Article ID 857984, 21 pages

http://dx.doi.org/10.1155/2013/857984

Review Article

## Properties of the Parabolic Anderson Model and the Anderson Polymer Model

Mathematics Department, University of California, Irvine, CA 92697-3875, USA

Received 3 September 2012; Accepted 7 November 2012

Academic Editors: M. Campanino and S. Sagitov

Copyright © 2013 Michael Cranston. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Linked References

- C. Henley and D. Huse, “Pinning and roughening of domain walls in ising systems,”
*Physical Review Letters*, vol. 54, no. 5, pp. 2708–2711, 1985. View at Google Scholar - F. Spitzer,
*Principles of Random Walk*, Springer-Verlag, New York, NY, USA, 1976. - D. A. Dawson and E. A. Perkins, “Long-time behavior and coexistence in a mutually catalytic branching model,”
*The Annals of Probability*, vol. 26, no. 3, pp. 1088–1138, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - A.-L. Barabási and H. E. Stanley,
*Fractal Concepts in Surface Growth*, Cambridge University Press, Cambridge, UK, 1995. View at Publisher · View at Google Scholar - R. A. Carmona, S. A. Grishin, and S. A. Molchanov, “Massively parallel simulations of motions in a Gaussian velocity field,” in
*Stochastic Modelling in Physical Oceanography*, vol. 39 of*Progress in Probability*, pp. 47–68, Birkhäuser Boston, Boston, Mass, USA, 1996. View at Google Scholar · View at Zentralblatt MATH - S. Molchanov, “Topics in statistical oceanography,” in
*Stochastic Modelling in Physical Oceanography*, vol. 39, pp. 343–380, Birkhäuser, Boston, Mass, USA, 1996. View at Google Scholar · View at Zentralblatt MATH - B. Bailey,
*Fast Dynamos. Theory of Solar and Planetary Dynamos*, NATO ASI, ed M.R.E. Proctor, University of Cambridge Press, Cambridge, UK, 1993. - P. K. S. Dittrikh, S. A. Molchanov, A. A. Ruzmaĭkin, and D. D. Sokolov, “Stationary distribution of the value of the magnetic field in a random flow,”
*Magnitnaya Gidrodinamika*, no. 3, pp. 9–12, 1988. View at Google Scholar - S. A. Molchanov and A. Ruzmaikin, “Lyapunov exponents and distributions of magnetic fields in dynamo models,” in
*The Dynkin Festschrift*, vol. 34 of*Progress in Probability*, pp. 287–306, Birkhäuser, Boston, Mass, USA, 1994. View at Google Scholar · View at Zentralblatt MATH - S. A. Molchanov, A. Ruzmaikin, and D. Sokolov, “Short-correlated random flow as a fast dynamo,”
*Soviet Physics—Doklady*, vol. 103, pp. 121–126, 1986. View at Google Scholar - S. A. Molchanov, A. Ruzmaikin, D. Sokolov, and Y. Zeldovich,
*Intermittency, Diffusion and Generation in Non-Stationary Random Medium*, vol. 7 of*Mathematical Physics Review*, Harwood Academic, New York, NY, USA, 1988. - M. Biskup and W. König, “Long-time tails in the parabolic Anderson model with bounded potential,”
*The Annals of Probability*, vol. 29, no. 2, pp. 636–682, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - J. Gärtner, W. König, and S. Molchanov, “Geometric characterization of intermittency in the parabolic Anderson model,”
*The Annals of Probability*, vol. 35, no. 2, pp. 439–499, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - J. Gärtner and S. A. Molchanov, “Parabolic problems for the Anderson model. I. Intermittency and related topics,”
*Communications in Mathematical Physics*, vol. 132, no. 3, pp. 613–655, 1990. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - R. van der Hofstad, W. König, and P. Mörters, “The universality classes in the parabolic Anderson model,”
*Communications in Mathematical Physics*, vol. 267, no. 2, pp. 307–353, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - W. König, “The parabolic Anderson model and its universality classes,” in
*Probability and Mathematical Physics*, vol. 42 of*CRM Proceedings and Lecture Notes*, pp. 283–298, American Mathematical Society, Providence, RI, USA, 2007. View at Google Scholar · View at Zentralblatt MATH - W. König, H. Lacoin, P. Mörters, and N. Sidorova, “A two cities theorem for the parabolic Anderson model,”
*The Annals of Probability*, vol. 37, no. 1, pp. 347–392, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - N. Ikeda and S. Watanabe,
*Stochastic Differential Equations and Diffusion Processes*, North Holland, 1981. - R. A. Carmona and S. A. Molchanov, “Parabolic Anderson problem and intermittency,”
*Memoirs of the American Mathematical Society*, vol. 108, no. 518, p. 125, 1994. View at Google Scholar · View at Zentralblatt MATH - A. M. Etheridge and K. Fleischmann, “Compact interface property for symbiotic branching,”
*Stochastic Processes and their Applications*, vol. 114, no. 1, pp. 127–160, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - F. Aurzada and L. Döring, “Intermittency and ageing for the symbiotic branching model,”
*Annales de l'Institut Henri Poincaré*, vol. 47, no. 2, pp. 376–394, 2011. View at Google Scholar - J. Blath, L. Döring, and A. Etheridge, “On the moments and the interface of the symbiotic branching model,”
*The Annals of Probability*, vol. 39, no. 1, pp. 252–290, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - J. Blath and M. Ortgiese, “Properties of the interface of the symbiotic branching model,” preprint.
- D. A. Dawson, K. Fleischmann, L. Mytnik, E. A. Perkins, and J. Xiong, “Mutually catalytic branching in the plane: uniqueness,”
*Annales de l'Institut Henri Poincaré. Probabilités et Statistiques*, vol. 39, no. 1, pp. 135–191, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - L. Mytnik, “Uniqueness for a mutually catalytic branching model,”
*Probability Theory and Related Fields*, vol. 112, no. 2, pp. 245–253, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - T. Shiga, “Stepping stone models in population genetics and population dynamics,” in
*Stochastic Processes in Physics and Engineering (Bielefeld, 1986)*, vol. 42 of*Applications of Mathematics*, pp. 345–355, Reidel, Dordrecht, The Netherlands, 1988. View at Google Scholar · View at Zentralblatt MATH - J. Gärtner and F. den Hollander, “Intermittency in a catalytic random medium,”
*The Annals of Probability*, vol. 34, no. 6, pp. 2219–2287, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - J. Gartner, F. den Hollander, and G. Maillard, “Quenched Lyapunov exponent for the parabolic Anderson model in a dynamic random environment,”
*Probability in Complex Physical Systems*, vol. 11, pp. 159–193, 2012. View at Google Scholar - J. Gärtner, F. den Hollander, and G. Maillard, “Intermittency of catalysts: voter model,”
*The Annals of Probability*, vol. 38, no. 5, pp. 2066–2102, 2010. View at Publisher · View at Google Scholar - J. Gartner, F. den Hollander, and G. Maillard, “Intermittency on catalysts: threedimensional symmetric exclusion,”
*Electronic Journal of Probability*, vol. 14, no. 72, pp. 2091–2129, 2009. View at Google Scholar - J. Gärtner, F. den Hollander, and G. Maillard, “Intermittency on catalysts: symmetric exclusion,”
*Electronic Journal of Probability*, vol. 12, no. 18, pp. 516–573, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - A. Greven and F. den Hollander, “Phase transitions for the long-time behavior of interacting diffusions,”
*The Annals of Probability*, vol. 35, no. 4, pp. 1250–1306, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - M. Birkner, “A condition for weak disorder for directed polymers in random environment,”
*Electronic Communications in Probability*, vol. 9, pp. 22–25, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - M. Cranston and S. A. Molchanov, “On phase transitions and limit theorems for homopolymers,” in
*Probability and Mathematical Physics*, vol. 42 of*CRM Proceedings and Lecture Notes*, pp. 97–112, American Mathematical Society, Providence, RI, USA, 2007. View at Google Scholar · View at Zentralblatt MATH - J. D. Esary, F. Proschan, and D. W. Walkup, “Association of random variables, with applications,”
*Annals of Mathematical Statistics*, vol. 38, pp. 1466–1474, 1967. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - C. M. Newman, “Normal fluctuations and the FKG inequalities,”
*Communications in Mathematical Physics*, vol. 74, no. 2, pp. 119–128, 1980. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - L. D. Pitt, “Positively correlated normal variables are associated,”
*The Annals of Probability*, vol. 10, no. 2, pp. 496–499, 1982. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - T. M. Liggett, “An improved subadditive ergodic theorem,”
*The Annals of Probability*, vol. 13, no. 4, pp. 1279–1285, 1985. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - M. Cranston, T. S. Mountford, and T. Shiga, “Lyapunov exponents for the parabolic Anderson model,”
*Acta Mathematica Universitatis Comenianae*, vol. 71, no. 2, pp. 163–188, 2002. View at Google Scholar · View at Zentralblatt MATH - C. Bezuidenhout and G. Grimmett, “The critical contact process dies out,”
*The Annals of Probability*, vol. 18, no. 4, pp. 1462–1482, 1990. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - R. Durrett, “Oriented percolation in two dimensions,”
*The Annals of Probability*, vol. 12, no. 4, pp. 999–1040, 1984. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - R. .A Carmona, L. Koralov, and S. Molchanov, “Asymptotics for the almost sure Lyapunov exponent for the solution of the parabolic Anderson problem,”
*Random Operators and Stochastic Equations*, vol. 9, no. 1, pp. 77–86, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - R. J. Adler,
*An Introduction to Continuity, Extrema, and Related Topics for General Gaussian Processes*, vol. 12 of*Lecture Notes Monograph Series*, Institute of Mathematical Statistics, Hayward, Calif, USA, 1990. View at Zentralblatt MATH - K. Johansson, “Transversal fluctuations for increasing subsequences on the plane,”
*Probability Theory and Related Fields*, vol. 116, no. 4, pp. 445–456, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - C. Licea, C. M. Newman, and M. S. T. Piza, “Superdiffusivity in first-passage percolation,”
*Probability Theory and Related Fields*, vol. 106, no. 4, pp. 559–591, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - T. Furuoya and T. Shiga, “Sample Lyapunov exponent for a class of linear Markovian systems over ${Z}^{d}$,”
*Osaka Journal of Mathematics*, vol. 35, no. 1, pp. 35–72, 1998. View at Google Scholar · View at Zentralblatt MATH - T. Shiga, “Ergodic theorems and exponential decay of sample paths for certain interacting diffusion systems,”
*Osaka Journal of Mathematics*, vol. 29, no. 4, pp. 789–807, 1992. View at Google Scholar · View at Zentralblatt MATH - T. Shiga and A. Shimizu, “Infinite-dimensional stochastic differential equations and their applications,”
*Journal of Mathematics of Kyoto University*, vol. 20, no. 3, pp. 395–416, 1980. View at Google Scholar · View at Zentralblatt MATH - J. T. Cox and A. Greven, “Ergodic theorems for infinite systems of locally interacting diffusions,”
*The Annals of Probability*, vol. 22, no. 2, pp. 833–853, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - T. M. Liggett,
*Interacting Particle Systems*, Classics in Mathematics, Springer, Berlin, Germany, 2005. - T. M. Liggett,
*Stochastic Interacting Systems: Contact, Voter and Exclusion Processes. Grundlehren der Mathematischen Wissenschaften*, vol. 324 of*Funda Principles of Mathematical Sciences*, Springer, Berlin, Germany, 1999. - M. Cranston and S. Molchanov, “Limit laws for sums of products of exponentials of iid random variables,”
*Israel Journal of Mathematics*, vol. 148, pp. 115–136, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - G. Ben Arous, L. V. Bogachev, and S. A. Molchanov, “Limit theorems for sums of random exponentials,”
*Probability Theory and Related Fields*, vol. 132, no. 4, pp. 579–612, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - G. Ben Arous, S. Molchanov, and A. F. Ramírez, “Transition from the annealed to the quenched asymptotics for a random walk on random obstacles,”
*The Annals of Probability*, vol. 33, no. 6, pp. 2149–2187, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - M. Cranston and S. Molchanov, “Quenched to annealed transition in the parabolic Anderson problem,”
*Probability Theory and Related Fields*, vol. 138, no. 1-2, pp. 177–193, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - G. Grimmett and H. Kesten, “First-passage percolation, network flows and electrical resistances,”
*Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete*, vol. 66, no. 3, pp. 335–366, 1984. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - J. T. Cox and G. Grimmett, “Central limit theorems for associated random variables and the percolation model,”
*The Annals of Probability*, vol. 12, no. 2, pp. 514–528, 1984. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - R. M. Balan, “A strong invariance principle for associated random fields,”
*The Annals of Probability*, vol. 33, no. 2, pp. 823–840, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - T.-S. Kim, “The invariance principle for associated random fields,”
*The Rocky Mountain Journal of Mathematics*, vol. 26, no. 4, pp. 1443–1454, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - C. M. Newman and A. L. Wright, “An invariance principle for certain dependent sequences,”
*The Annals of Probability*, vol. 9, no. 4, pp. 671–675, 1981. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - A. R. Dabrowski and A. Jakubowski, “Stable limits for associated random variables,”
*The Annals of Probability*, vol. 22, no. 1, pp. 1–16, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - P. Lèvy,
*Problèmes Complets D’analyse Fonctionelle*, Gauthiers-Villars, 1951. - M. Talagrand, “Concentration of measure and isoperimetric inequalities in product spaces,”
*Institut des Hautes Études Scientifiques*, no. 81, pp. 73–205, 1995. View at Google Scholar · View at Zentralblatt MATH - M. Talagrand, “New concentration inequalities in product spaces,”
*Inventiones Mathematicae*, vol. 126, no. 3, pp. 505–563, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - M. Talagrand, “A new look at independence,”
*The Annals of Probability*, vol. 24, no. 1, pp. 1–34, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - M. Ledoux,
*The Concentration of Measure Phenomenon*, vol. 89 of*Mathematical Surveys and Monographs*, American Mathematical Society, 2001. - A. Toubol, “High temperature regime for a multidimensional Sherrington-Kirkpatrick model of spin glass,”
*Probability Theory and Related Fields*, vol. 110, no. 4, pp. 497–534, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - C. Rovira and S. Tindel, “On the Brownian-directed polymer in a Gaussian random environment,”
*Journal of Functional Analysis*, vol. 222, no. 1, pp. 178–201, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - D. Nualart,
*The Malliavin Calculus and Related Topics*, Springer, Berlin, Germany, 2nd edition, 2006. - A. S. Üstünel and M. Zakai,
*Transformation of Measure on Wiener Space*, Springer, Berlin, Germany, 2000. - I. Ben-Ari, “Large deviations for partition functions of directed polymers in an IID field,”
*Annales de l'Institut Henri Poincaré Probabilités et Statistiques*, vol. 45, no. 3, pp. 770–792, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - M. Cranston, D. Gauthier, and T. S. Mountford, “On large deviation regimes for random media models,”
*The Annals of Applied Probability*, vol. 19, no. 2, pp. 826–862, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - M. Cranston, D. Gauthier, and T. S. Mountford, “On large deviations for the parabolic Anderson model,”
*Probability Theory and Related Fields*, vol. 147, no. 1-2, pp. 349–378, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - Y. Chow and Y. Zhang, “Large deviations in first-passage percolation,”
*The Annals of Applied Probability*, vol. 13, no. 4, pp. 1601–1614, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - J.-D. Deuschel and O. Zeitouni, “On increasing subsequences of I.I.D. Samples,”
*Combinatorics, Probability and Computing*, vol. 8, no. 3, pp. 247–263, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - G. Grimmett,
*Percolation*, Springer, New York, NY, USA, 1999. - J. M. Hammersley and D. J. A. Welsh, “First-passage percolation, subadditive processes, stochastic networks and generalized renewal theory,” in
*Bernoulli, Bayes*, J. Neyman and L. Le Cam, Eds., Laplace Anniversary Volume, Springer, Berlin, Germany, 1965. View at Google Scholar - R. A. Carmona and F. G. Viens, “Almost-sure exponential behavior of a stochastic Anderson model with continuous space parameter,”
*Stochastics and Stochastics Reports*, vol. 62, no. 3-4, pp. 251–273, 1998. View at Google Scholar · View at Zentralblatt MATH - M. Cranston and T. S. Mountford, “Lyapunov exponent for the parabolic Anderson model in ${R}^{d}$,”
*Journal of Functional Analysis*, vol. 236, no. 1, pp. 78–119, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - M. Rael, “Asymptotics of the Lyapunov exponent for the continuum parabolic Anderson model,” preprint.
- G. Amir, I. Corwin, and J. Quastel, “Probability distribution of the free energy of the continuum directed random polymer in $1+1$ dimensions,”
*Communications on Pure and Applied Mathematics*, vol. 64, no. 4, pp. 466–537, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - A. Auffinger and O. Louidor, “Directed polymers in random environment with heavy tails,”
*Communications on Pure and Applied Mathematics*. In press. - A. Cadel, S. Tindel, and F. Viens, “Sharp asymptotics for the partition function of some continuous-time directed polymers,”
*Potential Analysis*, vol. 29, no. 2, pp. 139–166, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - P. Carmona and Y. Hu, “On the partition function of a directed polymer in a Gaussian random environment,”
*Probability Theory and Related Fields*, vol. 124, no. 3, pp. 431–457, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - P. Carmona and Y. Hu, “Strong disorder implies strong localization for directed polymers in a random environment,”
*Latin American Journal of Probability and Mathematical Statistics*, vol. 2, pp. 217–229, 2006. View at Google Scholar · View at Zentralblatt MATH - F. Comets, T. Shiga, and N. Yoshida, “Directed polymers in a random environment: path localization and strong disorder,”
*Bernoulli*, vol. 9, no. 4, pp. 705–723, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - F. Comets, T. Shiga, and N. Yoshida, “Probabilistic analysis of directed polymers in a random environment: a review,” in
*Stochastic Analysis on Large Scale Interacting Systems*, vol. 39 of*Advanced Studies in Pure Mathematics.*, pp. 115–142, Mathematics Society, Tokyo, Japan, 2004. View at Google Scholar · View at Zentralblatt MATH - F. Comets and N. Yoshida, “Directed polymers in random environment are diffusive at weak disorder,”
*The Annals of Probability*, vol. 34, no. 5, pp. 1746–1770, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - G. Giacomin,
*Random Polymer Models*, Imperial College Press, 2007. View at Publisher · View at Google Scholar - H. Lacoin, “New bounds for the free energy of directed polymers in dimension $1+1$ and $1+2$,”
*Communications in Mathematical Physics*, vol. 294, no. 2, pp. 471–503, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - D. Marquez-Carreras, R. Rovira, and S. Tindel, “A model of continuous time polymer on the lattice,” In press, http://arxiv.org/abs/0802.3296.
- G. Moreno Flores, “Asymmetric directed polymers in random environments,” In press, http://arxiv.org/abs/1009.5576.
- G. Moreno Flores, J. Quastel, and D. Remenik, “Endpoint distribution of directed polymers in 1+1 dimensions,” Preprint, http://arxiv.org/abs/1106.2716.
- J. Moriarty and N. O'Connell, “On the free energy of a directed polymer in a Brownian environment,”
*Markov Processes and Related Fields*, vol. 13, no. 2, pp. 251–266, 2007. View at Google Scholar · View at Zentralblatt MATH - V. Vargas, “Strong localization and macroscopic atoms for directed polymers,”
*Probability Theory and Related Fields*, vol. 138, no. 3-4, pp. 391–410, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - E. Bolthausen, “A note on the diffusion of directed polymers in a random environment,”
*Communications in Mathematical Physics*, vol. 123, no. 4, pp. 529–534, 1989. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - J. Z. Imbrie and T. Spencer, “Diffusion of directed polymers in a random environment,”
*Journal of Statistical Physics*, vol. 52, no. 3-4, pp. 609–626, 1988. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - R. Song and X. Y. Zhou, “A remark on diffusion of directed polymers in random environments,”
*Journal of Statistical Physics*, vol. 85, no. 1-2, pp. 277–289, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - F. Comets and M. Cranston, “Overlaps and pathwise localization in the anderson polymer model,” http://arxiv.org/abs/1107.2011.
- F. Guerra, “Sum rules for the free energy in the mean field spin glass model,” in
*Mathematical Physics in Mathematics and Physics (Siena, 2000)*, vol. 30 of*Fields Institute Communications*, pp. 161–170, American Mathematical Society, Providence, RI, USA, 2001. View at Google Scholar · View at Zentralblatt MATH - F. Guerra and F. L. Toninelli, “The thermodynamic limit in mean field spin glass models,”
*Communications in Mathematical Physics*, vol. 230, no. 1, pp. 71–79, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - M. Talagrand,
*Spin Glasses: A Challenge for Mathematicians. Cavity and Mean Field Models*, Springer, Berlin, Germany, 2003. - M. Talagrand, “The Parisi formula,”
*Annals of Mathematics*, vol. 163, no. 1, pp. 221–263, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - C. Newman,
*Topics in Disordered Systems. Lectures in Mathematics ETH Zurich*, Birkhäuser, Basel, Switzerland, 1997.