P\'olya urn with memory kernel and asymptotic behaviours of autocorrelation function

Polya urn is a successive random drawing process of a ball from an urn that contains red and blue balls, in which the ball is returned to the urn with an additional ball of the same colour. The probability to draw a red ball is equal to the ratio of the red balls in the urn. We introduce arbitrary memory kernels to modify the probability. When the memory kernel decays by a power-law, there occurs a phase transition in the asymptotic behavior of the autocorrelation function. We introduce an auxiliary field variable that obeys a stochastic differential equation with a common noise. We estimate the power-law exponents of the leading and sub-leading terms of the autocorrelation function. We show that the power-law exponents changes discontinuously at the critical point.