We consider Reinforced Random Walks where transitions probabilities are a function of the proportions of times the walk has traversed an edge. We give conditions for recurrence or transience. A phase transition is observed, similar to Pemantle [7] on trees
References
Benaïm, Michel. Dynamics of stochastic approximation algorithms. Séminaire de Probabilités, XXXIII, 1–68, Lecture Notes in Math., 1709, Springer, Berlin, 1999.Benaïm, Michel. Dynamics of stochastic approximation algorithms. Séminaire de Probabilités, XXXIII, 1–68, Lecture Notes in Math., 1709, Springer, Berlin, 1999.
Benaim, Michel; Hirsch, Morris W. Dynamics of Morse-Smale urn processes. Ergodic Theory Dynam. Systems 15 (1995), no. 6, 1005–1030. 0846.60054 10.1017/S0143385700009767Benaim, Michel; Hirsch, Morris W. Dynamics of Morse-Smale urn processes. Ergodic Theory Dynam. Systems 15 (1995), no. 6, 1005–1030. 0846.60054 10.1017/S0143385700009767
Duflo, Marie. Algorithmes stochastiques.(French) [Stochastic algorithms] Mathématiques & Applications (Berlin) [Mathematics & Applications], 23. Springer-Verlag, Berlin, 1996. xiv+319 pp. ISBN: 3-540-60699-8Duflo, Marie. Algorithmes stochastiques.(French) [Stochastic algorithms] Mathématiques & Applications (Berlin) [Mathematics & Applications], 23. Springer-Verlag, Berlin, 1996. xiv+319 pp. ISBN: 3-540-60699-8
Hill B., Lane D., and Sudderth W.: A strong law for some generalized urn processes, Ann. Probab. 8, (1980), 214–226. MR566589 0429.60021 10.1214/aop/1176994772 euclid.aop/1176994772Hill B., Lane D., and Sudderth W.: A strong law for some generalized urn processes, Ann. Probab. 8, (1980), 214–226. MR566589 0429.60021 10.1214/aop/1176994772 euclid.aop/1176994772
Kosygina, Elena; Zerner, Martin P. W. Positively and negatively excited random walks on integers, with branching processes. Electron. J. Probab. 13 (2008), no. 64, 1952–1979. 1191.60113 10.1214/EJP.v13-572Kosygina, Elena; Zerner, Martin P. W. Positively and negatively excited random walks on integers, with branching processes. Electron. J. Probab. 13 (2008), no. 64, 1952–1979. 1191.60113 10.1214/EJP.v13-572
Pemantle, Robin. Phase transition in reinforced random walk and RWRE on trees. Ann. Probab. 16 (1988), no. 3, 1229–1241. MR942765 0648.60077 10.1214/aop/1176991687 euclid.aop/1176991687Pemantle, Robin. Phase transition in reinforced random walk and RWRE on trees. Ann. Probab. 16 (1988), no. 3, 1229–1241. MR942765 0648.60077 10.1214/aop/1176991687 euclid.aop/1176991687
Pemantle, Robin. Nonconvergence to unstable points in urn models and stochastic approximations. Ann. Probab. 18 (1990), no. 2, 698–712. 0709.60054 10.1214/aop/1176990853 euclid.aop/1176990853Pemantle, Robin. Nonconvergence to unstable points in urn models and stochastic approximations. Ann. Probab. 18 (1990), no. 2, 698–712. 0709.60054 10.1214/aop/1176990853 euclid.aop/1176990853
Pemantle, Robin. When are touchpoints limits for generalized Pólya urns? Proc. Amer. Math. Soc. 113 (1991), no. 1, 235–243. MR1055778 0725.60068Pemantle, Robin. When are touchpoints limits for generalized Pólya urns? Proc. Amer. Math. Soc. 113 (1991), no. 1, 235–243. MR1055778 0725.60068
Pemantle, Robin. A survey of random processes with reinforcement. Probab. Surv. 4 (2007), 1–79 (electronic). 1189.60138 10.1214/07-PS094Pemantle, Robin. A survey of random processes with reinforcement. Probab. Surv. 4 (2007), 1–79 (electronic). 1189.60138 10.1214/07-PS094
Williams, David. Probability with martingales.Cambridge Mathematical Textbooks. Cambridge University Press, Cambridge, 1991. xvi+251 pp. ISBN: 0-521-40455-X; 0-521-40605-6Williams, David. Probability with martingales.Cambridge Mathematical Textbooks. Cambridge University Press, Cambridge, 1991. xvi+251 pp. ISBN: 0-521-40455-X; 0-521-40605-6
Zerner, Martin P. W. Multi-excited random walks on integers. Probab. Theory Related Fields 133 (2005), no. 1, 98–122. 1076.60088 10.1007/s00440-004-0417-0Zerner, Martin P. W. Multi-excited random walks on integers. Probab. Theory Related Fields 133 (2005), no. 1, 98–122. 1076.60088 10.1007/s00440-004-0417-0
Zerner, Martin P. W. Recurrence and transience of excited random walks on $\Bbb Z\sp d$ and strips. Electron. Comm. Probab. 11 (2006), 118–128 (electronic). 1112.60086 10.1214/ECP.v11-1200Zerner, Martin P. W. Recurrence and transience of excited random walks on $\Bbb Z\sp d$ and strips. Electron. Comm. Probab. 11 (2006), 118–128 (electronic). 1112.60086 10.1214/ECP.v11-1200