Markov chains and random walks

• see also Stochastic processes (mathematical), Fluctuation phenomena and stochastic processes (in physics)
• Markov chains
• Graphs
• Geometrically disordered lattices
• Lattices
• One-dimensional systems
• Fractals
• Small world networks
• Continuous systems
• One-dimensional continuous systems
• Specific models
• see also Combinatorics, Difference equations, Matrix theory, Random matrix theory, Random fields
• see also Disordered structures, Polymers, Diffusion-limited reactions

Random walk (general studies)

I Eliazar, Regular and anomalous diffusion: I. Foundations, JPA 57, 233002 (2024) – doi

R Metzler, J Jeon, A Cherstvy, E Barkai, Anomalous diffusion models and their properties: non-stationarity, non-ergodicity, and ageing at the centenary of single particle tracking, PCCP 16, 24128 (2014) – doi

J Rudnick, G Gaspari, Elements of the random walk: An introduction for advanced students and researchers (CUP, 2004)

S Redner, A guide to first-passage processes (CUP, 2001) – err, exercises

M Bazant, Lecture notes on random walks (2001) – ps

D Ben-Avraham, S Havlin, Diffusion and reactions in fractals and disordered systems (CUP, 2000)

B D Hughes, Random walks and random environments. Volume 1: Random walks (Clarendon Press, Oxford, 1995)

B D Hughes, Random walks and random environments. Volume 2: Random environments (Clarendon Press, Oxford, 1996)

G H Weiss, Aspects and applications of the random walk (North­Holland, Amsterdam, 1994)

J Bouchaud, A Georges, Anomalous diffusion in disordered media: Statistical mechanics, models and physical applications, PR 195, 127 (1990)

S Havlin, D Ben-Avraham, Diffusion in disordered media, AP 36, 695 (1987)

J W Haus, K W Kehr, Diffusion in regular and disordered lattices, PR 150, 263 (1987)

Percolation (general studies)

Percolation threshold (Wikipedia)

A Hunt, Percolation theory for flow in porous media, LNP 674 (Springer, 2005)

Тарасевич Ю Ю, Перколяция: теория, приложения, алгоритмы (УРСС, 2002)

R Romer, Percolation, renormalization and the Quantum-Hall transition, cond-mat/0106004

G Grimmett, Percolation, in Development of mathematics 1950-2000, ed Pier J-P, p 547 (Birkhauser, 2000)

G Grimmett, Percolation (Springer, New York, 1999)

A Bain, The Percolation Phase Transition (Lecture Notes, 1999)

H Stanley, J Andrade, S Havlin, H Makse, B Suki, Percolation phenomena: a broad-brush introduction with some recent applications to porous media, liquid water, and city growth, PA 266, 5 (1999)

G Grimmett, Percolation and disordered system, LNM 1665, 153 (1997)

D Stauffer, A Aharony, Introduction to percolation theory (TF, 1994)

Кестен Х, Теория просачивания для математиков (Мир, 1986)

Соколов И М, Размерности и другие геометрические критические показатели в теории протекания, УФН 150, 221 (1986)

J Essam, Percolation theory, RPP 43, 833 (1980)

D Stauffer, Scaling theory of percolation clusters, PR 54, 1 (1979)

Шкловский Б И, Эфрос А Л, Теория протекания и проводимость сильно неоднородных сред, УФН 117, 401 (1975)

V Shante, S Kirkpatrick, An introduction to percolation theory, AP 20, 325 (1971)

C Domb, Series expansions for ferromagnetic models, AP 19, 339 (1970)

Miscellaneous

Anomalous transport: Foundations and Applications, ed Radons G, Klages R, Sokolov I M (Wiley, 2008)

M Sahimi, Non-linear and non-local transport processes in heterogeneous media: from long-range correlated percolation to fracture and materials breakdown, PR 306, 213 (1998)

Anomalous diffusion: From basics to applications, ed A Pekalski, K Sznajd-Weron, LNP 519 (Springer, 1999)

Fractals and disordered systems, ed Bunde A, S Havlin (Springer, 1996)

Contemporary problems in statistical physics, ed Weiss G H (SIAM, Philadelphia, 1994)

P Revesz, Random walks of infinitely many particles (WSPC, 1994)

E Montroll, M Shlesinger, On the wonderful world of random walks, in Nonequilibrium Phenomena II, ed Montroll E W, Lebowitz J L, p 1 (Elsevier, 1984)