Phases in the two-color tenable zero-balanced Pólya process

The Pólya process is obtained by embedding the usual (discrete-time) Pólya urn scheme in continuous time. We study the class of tenable Pólya processes of white and blue balls with zero balance (no change in n, the total number of balls, over time). This class includes the (continuous-time) Ehrenfest process and the (continuous-time) Coupon Collector's process. We look at the composition of the urn at time tn (dependent on n). We identify a critical phase of tn at the edges of which phase transitions occur. In the subcritical phase, under proper scaling the number of white balls is concentrated around a constant. In the critical phase, we have sufficient variability for an asymptotic normal distribution to be in effect. In this phase, the influence of the initial conditions is still somewhat pronounced. Beyond the critical phase, the urn is very well mixed with an asymptotic normal distribution, in which all initial conditions wither away. The results are obtained by an analytic approach utilizing partial differential equations.