Home | Current | Past volumes | About | Login | Notify | Contact | Search
 Probability Surveys > Vol. 4 (2007) open journal systems 


A survey of random processes with reinforcement

Robin Pemantle, University of Pennsylvania


Abstract
The models surveyed include generalized Polya urns, reinforced random walks, interacting urn models, and continuous reinforced processes. Emphasis is on methods and results, with sketches provided of some proofs. Applications are discussed in statistics, biology, economics and a number of other areas.

AMS 2000 subject classifications: Primary 60J20, 60G50; secondary 37A50.

Keywords: urn model, urn scheme, Pólya’s urn, stochastic approximation, dynamical system, exchangeability, Lyapunov function, reinforced random walk, ERRW, VRRW, learning, agent-based model, evolutionary game theory, self-avoiding walk.

Creative Common LOGO

Full Text: PDF


Pemantle, Robin, A survey of random processes with reinforcement, Probability Surveys, 4, (2007), 1-79 (electronic).

References

[AEK83]     B. Arthur, Y. Ermoliev, and Y. Kaniovskii. A generalized urn problem and its applications. Cybernetics, 19:61–71, 1983.

[AEK87]     B. Arthur, Y. Ermoliev, and Y. Kaniovskii. Path dependent processes and the emergence of macro-structure. Eur. J. Oper. Res., 30:294–303, 1987.

[AHS05]     J. Alford, J. Hibbing, and K. Smith. The challenge evolutionary biology poses for rational choice. Paper presented at the annual meeting of the APSA, 2005.

[AK68]     K. Athreya and S. Karlin. Embedding of urn schemes into continuous time Markov branching processes and related limit theorems. Ann. Math. Statist., 39:1801–1817, 1968. MR0232455

[Ald90]     D. Aldous. The random walk construction of uniform spanning trees and uniform labelled trees. SIAM J. Disc. Math., 3:450–465, 1990. MR1069105

[Ale05]     J. Alexander. Artificial virtue: the structural evolution of morality. Preprint, 2005.

[APP83]     D. Amit, G. Parisi, and L. Peliti. Asymptotic behavior of the ‘true’ self avoiding walk. Phys. Rev. B, 27:1635–1645, 1983. MR0690540

[Art90]     B. Arthur. Positive feedbacks in the economy. Scientific American, pages 92–99, 1990.

[Ath68]     K. Athreya. Some results on multitype continuous time Markov branching processes. Ann. Math. Statist., 38:347–357, 1968. MR0221600

[BA99]     A.-L. Barábasi and R. Albert. Emergence of scaling in random networks. Science, 286:509–512, 1999. MR2091634

[Bar81]     B. Barsky. The Beta-spline: a Local Representation Based on Shape Parameters and Fundamental Geometric Measures. Doctoral Dissertation. University of Utah, 1981.

[BB03]     M. Baracho and I. Baracho. An analysis of the spontaneous mutation rate measurement in filamentous fungi. Genetics and Molec. Biol., 26:83–87, 2003.

[BBA99]     A. Banerjee, P. Burlina, and F. Alajaji. Image segmentation and labeling using the Polya urn model. IEEE Transactions on Image Processing, 8:1243–1253, 1999.

[Ben93]     M. Benaïm. Sur la nature des ensembles limites des trajectoires des algorithmes d’approximation stochastiques de type Robbins-Monro. C. R. Acad. Sci. Paris Sér. I Math., 317:195–200, 1993. MR1231421

[Ben97]     M. Benaïm. Vertex-reinforced radnom walks and a conjecture of Pemantle. Ann. Probab., 25:361–392, 1997. MR1428513

[Ben99]     M. Benaïm. Dynamics of stochastic approximation algorithms. In Seminaires de Probabilités XXXIII, volume 1709 of Lecture notes in mathematics, pages 1–68. Springer-Verlag, Berlin, 1999. MR1767993

[Ben00]     M. Benaïm. Convergence with probability 1 of stochastic approximation algorithms whose average is cooperative. Nonlinearity, 13:601–616, 2000. MR1758990

[BH95]     M. Benaïm and M. Hirsch. Dynamics of Morse-Smale urn processes. Ergodic Theory Dynam. Systems, 15:1005–1030, 1995. MR1366305

[BH96]     M. Benaïm and M. Hirsch. Asymptotic pseudotrajectories and chain-recurrent flows, with applications. J. Dynam. Differential Equations, 8:141–176, 1996. MR1388167

[BH99a]     M. Benaïm and M. Hirsch. Mixed equilibria and dynamical systems arising from fictitious play in repeated games. Games and Econ. Beh., 29:36–72, 1999. MR1729309

[BH99b]     M. Benaïm and M. Hirsch. Stochastic approximation algorithms with constant step size whose average is cooperative. Ann. Appl. Probab., 9:216–241, 1999. MR1682576

[BHS05]     M. Benaïm, J. Hofbauer, and S. Sorin. Stochastic approximations and differential inclusions, I. SIAM Journal on Optimization and Control, 44:328–348, 2005. MR2177159

[BHS06]     M. Benaïm, J. Hofbauer, and S. Sorin. Stochastic approximations and differential inclusions, II. In Press, 2006. MR2177159

[BJK62]     R. Bradt, R. Johnson, and S. Karlin. On sequential designs for maximizing the sum of n observations. Ann. Math. Statist., 31:1060–1074, 1962. MR0087288

[BK64]     D. Blackwell and D. Kendall. The Martin boundary for Pólya’s urn scheme. Journal of Applied Probability, 1:284–296, 1964. MR0176518

[BL03]     P. Bonacich and T. Liggett. Asymptotics of a matrix valued Markov chain arising in sociology. Stochastic Process. Appl., 104:155–171, 2003. MR1956477

[BLR02]     M. Benaïm, M. Ledoux, and O. Raimond. Self-interacting diffusions. Prob. Theory Related Fields, 122:1–41, 2002. MR1883716

[BM55]     R. Bush and F. Mosteller. Stochastic Models for Learning. John Wiley, New York, 1955. MR0070143

[BM73]     D. Blackwell and J. McQueen. Ferguson distributions via Pólya urn schemes. Ann. Statist., 1:353–355, 1973. MR0362614

[BMP90]     A. Benveniste, M. Métivier, and P. Priouret. Stochastic Approximation and Adaptive Algorithms, volume 22 of Applications of Mathematics. Springer-Verlag, New York, 1990.

[Bon02]     E. Bonabeau. Agent-based modeling: Methods and techniques for simulating human systems. Proc. Nat. Acad. Sci. U.S.A., 99 (Supplement 3):7280–7287, 2002.

[Bow75]     R. Bowen. ω-limit sets of axiom A diffeomorphisms. J. Diff. Eq., 18:333–339, 1975. MR0413181

[BP85]     A. Bagchi and A. Pal. Asymptotic normality in the generalized Pólya-Eggenberger urn model, with an application to computer data structures. SIAM J. Alg. Disc. Meth., 6:394–405, 1985. MR0791169

[BR02]     M. Benaïm and O. Raimond. On self-attracting/repelling diffusions. Comptes Rendus Acad. Sci. Paris, ser. I, 335:541–544, 2002. MR1936828

[BR03]     M. Benaïm and O. Raimond. Self-interacting diffusions, II: convergence in law. Ann. Inst. H. Poincaré, prob. stat., 39:1043–1055, 2003. MR2010396

[BR05]     M. Benaïm and O. Raimond. Self-interacting diffusions, III: symmetric interactions. Ann. Probab., 33:1716–1759, 2005. MR2165577

[Bra98]     O. Brandière. Some pathological traps for stochastic approximation. SIAM J. on Control and Optimization, 36:1293–1314, 1998. MR1618037

[Bro51]     G. Brown. Iterative solutions of games by fictitious play. In T. C. Koopmans, editor, Activity Analysis of Production and Allocation. John Wiley & Sons, New York, 1951. MR0056265

[BRST01]     B. Bollobás, O. Riordan, J. Spencer, and G. Tusnády. The degree sequence of a scale-free random graph process. Random Structures and Algorithms, 18:279–290, 2001. MR1824277

[BS85]     A. Beretti and A. Sokal. New Monte Carlo method for the self-avoiding walk. J. Statist. Phys., 40:483–531, 1985. MR0806712

[BS02]     J. Busemeyer and J. Stout. A contribution of cognitive decision models to clinical assessment: decomposing performance on the Bechara Gambling Task. Psychological Assessment, 14:253–262, 2002.

[BST04]     M. Benaïm, S. Schreiber, and P. Tarrès. Generalized urn models of evoutionary processes. Ann. Appl. Prob, 14:1455–1478, 2004. MR2071430

[BW03]     I. Benjamini and D. Wilson. Excited random walk. Elec. Comm. Prob., 8:paper 9, 2003. MR1987097

[CD87]     D Coppersmith and P. Diaconis. Random walk with reinforcement. Unpublished manuscript, 1987.

[CL03]     F. Chung and L. Lu. Average distances in random graphs with given expected degrees. Internet Mathematics, 1:91–114, 2003. MR2076728

[CL06a]     F. Chung and L. Lu. Complex Graphs and Networks. CBMS Regional Conference Series in Mathematics. American Mathematical Society, Providence, 2006. MR2248695

[CL06b]     C. Cotar and V. Limic. Attraction time for strongly reinforced random walks. arXiv, math.PR/0612048:27, 2006.

[CLJ95]     M. Cranston and Y. Le Jan. Self-attracting diffusions: two case studies. Math. Ann., 303:87–93, 1995. MR1348356

[CM96]     M. Cranston and T. Mountford. The strong law of large numbers for a Brownian polymer. Ann. Probab., 24:1300–1323, 1996. MR1411496

[Coh76]     J. Cohen. Irreproducible results in the breeding of pigs. Bioscience, 26:241–245, 1976.

[Col04]     A. Collevecchio. Limit Theorems for Reinforced Random Walks on Trees. Doctoral Dissertation. Purdue University, 2004.

[Col06a]     A. Collevecchio. Limit theorems for reinforced random walks on certain trees. Prob. Theory Related Fields, 136:81–101, 2006. MR2240783

[Col06b]     A. Collevecchio. On the transience of processes defined on galton-watson trees. Ann. Probab., 34:870–878, 2006. MR2243872

[Con78]     C. Conley. Isolated Invariant Sets and the Morse Index, volume 38 of CBMS Regional Conference Series in Mathematics. American Mathematical Society, Providence, 1978. MR0511133

[CPY98]     R. Carmona, F. Petit, and M. Yor. Beta variables as time spent in [0,] by certain perturbed reflecting Brownian motions. J. London Math. Soc., 58:239–256, 1998. MR1670130

[Dav90]     B. Davis. Reinforced random walk. Prob. Theory Related Fields, 84:203–229, 1990. MR1030727

[Dav96]     B. Davis. Weak limits of perturbed random walks and the equation Y t = Bt + α sup{Y s : s t} + β inf{Y s : s t}. Ann. Probab., 24:2007–2023, 1996. MR1415238

[Dav99]     B. Davis. Reinforced and perturbed random walks. In Random Walks, volume 9 of Bolyai Soc. Math. Stud., pages 113–126. János bolyai Math. Soc., Budapest, 1999. MR1752892

[dF38]     B. de Finetti. Sur la question d’equivalence partielle. Actualites Scientifiques et Industrielles, 79, 1938.

[DF66]     L. Dubins and D. Freedman. Random distribution functions. Proc. Fifth Berkeley Symp. Math. Statist. Prob., 2:183–214, 1966. MR0214109

[DF80]     P. Diaconis and D. Freedman. de Finetti’s theorem for Markov chains. Ann. Probab., 8:115–130, 1980. MR0556418

[DGVEE05]     E. Di Giuseppe, D. Vento, C. Epifani, and S. Esposito. Analysis of dry and wet spells from 1870 to 2000 in four italian sites. Geophysical Research Abstracts, 7:6, 2005.

[Dia88]     P. Diaconis. Recent progress on de Finetti’s notion of exchangeability. In J. Bernardo, M. de Groot, D. Lindley, and A. Smith, editors, Bayesian Statistics, pages 111–125. Oxford University Press, Oxford, 1988. MR1008047

[Die05]     J. Die. A once edge-reinforced random walk on a Galton-Watson tree is transient. Statist. Probab. Lett., 73:115–124, 2005. MR2159246

[Dir00]     G. Dirienzo. Using urn models for the design of clinical trials. Indian Journal of Statistics, 62:43–69, 2000. MR1789790

[DKL02]     R. Durrett, H. Kesten, and V. Limic. Once edge-reinforced random walk on a tree. Prob. Theor. Rel. Fields, 122:567–592, 2002. MR1902191

[DR92]     R. Durrett and L. C. G. Rogers. Asymptotic behavior of Brownian polymers. Prob. Theor. Rel. Fields, 92:337–349, 1992. MR1165516

[DR06]     P. Diaconis and S. Rolles. Bayesian analysis for reversible Markov chains. Annals of Statistics, 34:1270–1292, 2006.

[Duf96]     M. Duflo. Algorithmes Stochastiques. Springer, Berlin, 1996. MR1612815

[Dur04]     R. Durrett. Probability: Theory and Examples. Duxbury Press, Belmont, CA, third edition, 2004. MR1609153

[DV97]     M. Drmota and V. Vatutin. Limiting distributions in branching processes with two types of particles. In Classical and modern branching processes, volume 84 of IMA volumes in Mathematics and Applications, pages 89–110. Springer, New York, 1997. MR1601709

[DV02]     B. Davis and S. Volkov. Continuous time vertex-reinforced jump processes. Prob. Theory Related Fields, 123:281–300, 2002. MR1900324

[DV04]     B. Davis and S. Volkov. Vertex-reinforced jump processes on trees and finite graphs. Prob. Theory Related Fields, 128:42–62, 2004. MR2027294

[EE07]     P. Ehrenfest and T. Ehrenfest. Über zwei bekannte Einwände gegen das Boltzmannsche H-theorem. Physikalische Zeitschrift, 8:311–314, 1907.

[EL04]     B. Eidelson and I. Lustick. Vir-pox: An agent-based analysis of smallpox preparedness and response policy. Journal of Artificial Societies and Social Simulation, 7, 2004.

[Ell93]     G. Ellison. Learning, local interaction, and coordination. Econometrica, 61:1047–1071, 1993. MR1234793

[EP23]     F. Eggenberger and G. Pólya. Über die Statistik vorketter vorgänge. Zeit. Angew. Math. Mech., 3:279–289, 1923.

[ES54]     W. Estes and J. Straughan. Analysis of a verbal conditioning situation in terms of statistical choice behavior under extended training with shifting probabilities of reinforcement. J. Experimental Psychology, 47:225–234, 1954.

[Fel62]     D. Feldman. Contributions to the “two-armed bandit” problem. Ann. Statist., 33:847–856, 1962. MR0145625

[Fel68]     W. Feller. An Introduction to Probability Theory and its Applications, vol. I. John Wiley & Sons, New York, third edition, 1968. MR0228020

[Fel71]     W. Feller. An Introduction to Probability Theory and its Applications, vol. II. John Wiley & Sons, New York, second edition, 1971. MR0270403

[Fer73]     T. Ferguson. A Bayesian analysis of some nonparamteric problems. Ann. Statist., 1:209–230, 1973. MR0350949

[Fer74]     T. Ferguson. Prior distributions on spaces of probability measures. Ann. Statist., 2:615–629, 1974. MR0438568

[FGP05]     P. Flajolet, J. Gabarró, and H. Pekari. Analytic urns. Ann. Probab., 33:1200–1233, 2005. MR2135318

[FK93]     D. Fudenberg and D. Kreps. Learning mixed equilibria. Games and Econ. Beh., 5:320–367, 1993. MR1227915

[Flo49]     P. Flory. The configuration of a real polymer chain. J. Chem. Phys., 17:303–310, 1949.

[FM02]     A. Flache and M. Macy. Stochastic collusion and the power law of learning. J. Conflict Res., 46:629–653, 2002.

[Fre65]     D. Freedman. Bernard Friedman’s urn. Ann. Math. Statist., 36:956–970, 1965. MR0177432

[Fri49]     B. Friedman. A simple urn model. Comm. Pure Appl. Math., 2:59–70, 1949. MR0030144

[FvZ70]     J. Fabius and W. van Zwet. Some remarks on the two-armed bandit. Ann. Math. Statist., 41:1906–1916, 1970. MR0278454

[Gol85]     R. Goldman. Pólya’s urn model and computer-aided geometric design. SIAM J. Alg. Disc. Meth., 6:1–28, 1985. MR0772172

[Gol88a]     R. Goldman. Urn models and beta-splines. Constructive approximation, 4:265–288, 1988. MR0940295

[Gol88b]     R. Goldman. Urn models, approximations and splines. J. Approx. Theory, 54:1–66, 1988. MR0951029

[Goo65]     I. J. Good. The Estimation of Probabilities: An Essay on Modern Bayesian Methods, volume 30 of Research Monographs. M.I.T. Press, Cambridge, MA, 1965. MR0185724

[Gre91]     D. Greenberg. Modeling criminal careers. Criminology, 29:17–46, 1991.

[GY20]     M. Greenwood and U. Yule. Inquiry into the nature of frequency distributions representative of multiple happenings with particular reference to the occurrence of multiple attacks of disease or repeated accidents. J. Royal. Stat. Soc., 83:255–279, 1920.

[Har73]     J. Harsanyi. Games with randomly disturbed payoffs: a new rationale for mixed-strategy equilibrium points. Int. J. Game Theory, 2:1–16, 1973. MR0323363

[Her70]     R. Herrnstein. On the law of effect. J. Anal. Exp. Behav., 13:243–266, 1970.

[HLS80]     B. Hill, D. Lane, and W. Sudderth. A strong law for some generalized urn processes. Ann. Probab., 8:214–226, 1980. MR0566589

[HM54]     J. Hammersley and K. Morton. Poor man’s Monte Carlo. J. Royal Stat. Soc. B, 16:23–38, 1954. MR0064475

[HS88]     J. Hofbauer and K. Sigmund. The Theory of Evolution and Dynamical Systems. Cambridge University Press, Cambridge, 1988. MR1071180

[HS98]     J. Hofbauer and K. Sigmund. Evolutionary Games and Population Dynamics. Cambridge University Press, Cambridge, 1998. MR1635735

[HS02]     J. Hofbauer and W. Sandholm. On the global convergence of stochastic fictitious play. Econometrica, 70:2265–2294, 2002. MR1939897

[INW66]     N. Ikeda, M. Nagasawa, and S. Watanabe. A construction of branching Markov processes. Proceedings of the Japan Academy, 42:380–384, 1966. MR0202198

[Jan82]     K. Janardhan. Correlation between the numbers of two types of children in a family using the Markov-Pólya model. Math. Biosci., 62:123–136, 1982. MR0684815

[Jan04]     S. Janson. Functional limit theorems for multitype branching processes and generalized Pólya urns. Stochastic Process. Appl., 110:177–245, 2004. MR2040966

[Jan05]     S. Janson. Limit theorems for triangular urn schemes. Prob. Theory Related Fields, 134:417–452, 2005. MR2226887

[Jor93]     J. Jordan. Three problems in learning mixed-strategy Nash equilibria. Games and Economic Behavior, 5:368–386, 1993. MR1227916

[KC78]     H. Kushner and D. Clark. Stochastic Approximation for Constrained and Unconstrained Systems, volume 26 of Applied Mathematical Sciences. Springer-Verlag, New York, 1978. MR0499560

[Kes63]     H. Kesten. On the number of self-avoiding walks. J. Math. Phys., 4:960–969, 1963. MR0152026

[Kin99]     J. F. C. Kingman. Martingales in the OK corral. Bull. London Math. Soc., 31:601–606, 1999. MR1703841

[KK01]     K. Khanin and R. Khanin. A probabilistic model for establishment of neuron polarity. J. Math. Biol., 42:26–40, 2001. MR1820779

[KKO+05]     S. Kakade, M. Kearns, L. Ortiz, R. Pemantle, and S. Suri. Economic properties of social networks. In L. Saul, Y. Weiss, and L. Bottou, editors, Proceedings of NIPS (2004), volume 17 of Advances in Neural Information Processing Systems. M. I. T. Press, Cambridge, MA, 2005.

[KMR93]     M. Kandori, G. Mailath, and R. Rob. Learning, mutation, and long run equilibria in games. Econometrica, 61:29–56, 1993. MR1201702

[KMR00]     S. Kotz, H. Mahmoud, and P. Robert. On generalized Pólya urn models. Stat. Prob. Let., 49:163–173, 2000. MR1790166

[KR99]     M. Keane and S. Rolles. Edge-reinforced random walk on finite graphs. In Infinite Dimensional Stochastic Analysis, volume 52 of Verhandelingen, Afdeling Natuurkunde. Eerste Reeks. Koninklijke Nederlandse Akademie van Wetenschappen. [Proceedings, Physics Section. Series 1. Royal Netherlands Academy of Arts and Sciences], pages 217–234. R. Neth. Acad. Arts Sci., Amsterdam, 1999. MR1832379

[KV03]     J. F. C. Kingman and S. Volkov. Solution to the OK Corral problem via decoupling of Friedman’s urn. J. Theoret. Probab., 16:267–276, 2003. MR1956831

[Law80]     G. Lawler. A self-avoiding random walk. Duke Math. J., 47:655–693, 1980. MR0587173

[Law91]     G Lawler. Intersections of Random Walks. Probability and its Applications. Birkhäuser, Boston, 1991. MR1117680

[Lim03]     V. Limic. Attracting edge property for a class of reinforced random walks. Ann. Probab., 31:1615–1654, 2003. MR1989445

[Lju77]     L. Ljung. Analysis of recursive stochastic algorithms. IEEE Transactions on Automatic Control, AC-22:551–575, 1977. MR0465458

[Löw23]     K. Löwner. Untersuchungen über schlichte konforme Abbildungen des Einheitskreises, I. Math. Ann., 89:103–121, 1923. MR1512136

[LPT04]     D. Lamberton, G. Pagès, and P. Tarrès. When can the two-armed bandit algorithm be trusted? Ann. Appl. Prob., 14:1424–1454, 2004. MR2071429

[LS97]     H. Levine and A. Sleeman. A system of reaction-diffusion equations arising in the theory of reinforced random walks. SIAM J. Appl. Math., 57:683–730, 1997. MR1450846

[LSW04]     G. Lawler, O. Schramm, and W. Werner. A self-avoiding random walk. Ann. Probab., 32:939–995, 2004. MR2044671

[LT06]     V. Limic and P. Tarrès. Attracting edge and strongly edge reinforced random walks. Preprint, page 25, 2006.

[Mah98]     H. Mahmoud. On rotations in fringe-balanced binary trees. Information Processing Letters, 65:41–46, 1998. MR1606251

[Mah03]     H. Mahmoud. Pólya urn models and connections to random trees: a review. J. Iranian. Stat. Soc., 2:53–114, 2003.

[Mah04]     H. Mahmoud. Pólya-type urn models with multiple drawings. J. Iranian Stat. Soc., 3:165–173, 2004.

[Min74]     H. Minikata. A geometrical aspect of multiplicity distribution and elastic diffraction scattering. Prg. Theor. Phys., 51:1481–1487, 1974.

[Mit03]     M. Mitzenmacher. A brief history of generative models for power law and lognormal distributions. Internet Mathematics, 1:226–251, 2003. MR2077227

[Miy61]     K. Miyasawa. On the convergence of the learning process in a 2×2 non-zero sum two-person game. Economic Research Program, Research Memorandom number 33, 1961.

[ML82]     D. Mackerro and H. Lawson. Weather limitations on the applications of dinoseb-in-oil for cane vigour control in raspberry. Ann. Appl. biol., 100:527–538, 1982.

[MR90]     P. Milgrom and J. Roberts. Rationalizability, learning, and equilibrium in games with strategic complementarities. Econometrica, 58:1255–1278, 1990. MR1080810

[MR05]     F. Merkl and S. Rolles. Edge-reinforced random walk on a ladder. Ann. Probab., 33:2051–2093, 2005. MR2184091

[MR06]     F. Merkl and S. Rolles. Linearly edge-reinforced random walks. In Dynamics & Stochastics: Festschrift in honor of M. S. Keane, volume 48 of IMS Lecture Notes – Monograph Series, pages 66–77. Institute of Mathematical Statistics Press, Hayward, CA, 2006.

[MR07]     F. Merkl and S. Rolles. A random environment for linearly edge-reinforced random walks on infinite graphs. Prob. Theor. Rel. Fields, To appear, 2007.

[MS74]     J. Maynard Smith. The theory of games and the evolution of animal conflicts. J. Theoret. Biol, 47:209–221, 1974. MR0444115

[MS82]     J. Maynard Smith. Evolution and the Theory of Games. Cambridge University Press, Cambridge, 1982.

[MS92]     H. Mahmoud and R. Smythe. Asymptotic joint normality of out-degrees of nodes in random recursive trees. Rand. Struc. Alg., 3:255–266, 1992. MR1164839

[MS93]     N. Madras and G. Slade. The Self-Avoiding Walk. Probability and its Applications. Birkhäuser, Boston, 1993. MR1197356

[MS95]     H. Mahmoud and R. Smythe. Probabilistic analysis of bucket recursive trees. Theor. Comp. Sci., 144:221–249, 1995. Home | Current | Past volumes | About | Login | Notify | Contact | Search

Probability Surveys. ISSN: 1549-5787