First passage properties of a generalized Pólya urn

Abstract

Abstract.A generalized two-component Pólya urn process, parameterized by a variable  , is studied in terms of the likelihood that due to fluctuations the initially smaller population in a scenario of competing population growth eventually becomes the larger, or is the larger after a certain passage of time. By casting the problem as an inhomogeneous directed random walk we quantify this role-reversal phenomenon through the first passage probability that equality in size is first reached at a given time, and the related exit probability that equality in size is reached no later than a given time. Using an embedding technique, exact results are obtained which complement existing results and provide new insights into behavioural changes (akin to phase transitions) which occur at defined values of  .Emails:

Random walks exhibiting anomalous diffusion: elephants, urns and the limits of normality

“…As well as providing a unifying perspective, the approximations used here to study the   T limit are controlled precisely and the analysis is consequently tighter than the Fokker-Planck approach. Similar findings are reported in different settings relating to random walks with slightly modified memory rules [4,11], bond percolation on random recursive trees [22], two-component urn processes [27,31,32] and voting models with two types of voter behaviour [37].…”

“…In other fields, the nature of urn-based random walk models has been explored in parallel [26][27][28][29][30][31][32]. We exploit this here by demonstrating asymptotic equivalence of Model I to a non-linear generalization of the standard twocomponent Pólya urn process [31], which we call Model II. This generalization has found application in many fields such as neuronal development [33], the organization of growing networks exhibiting preferential attachment [34][35][36], information cascade in voter models [37,38] and the emergence of macrostructure in economics [39,40].…”

“…Recent studies of the original ERW have benefited from a mapping onto an urn process, which has greatly aided understanding of the limiting behaviour as the number of steps grows large [16]. In other fields, the nature of urn-based random walk models has been explored in parallel [26][27][28][29][30][31][32]. We exploit this here by demonstrating asymptotic equivalence of Model I to a non-linear generalization of the standard twocomponent Pólya urn process [31], which we call Model II.…”

Recurrence time correlations in random walks with preferential relocation to visited places

“…Finally, in the last step we have rewritten the conditional distribution f n (t|m n−1 ) in terms of the conditional survival probability S n (t|m n−1 ). The reason for writing the mean recurrence time in that way is because for each recurrence process one can provide explicit expressions equivalent to (18) and (19) for the first-passage, respectively:…”

“…Note first of all that both expressions for the survival probabilities reduce to (18) and (19) for n = 1 (since m 0 = 0 and so the conditional probability is unnecessary in that case). Now, we observe that the memory mode explicitly contributes to the survival probability through the last term within the parenthesis of (26).…”