Recursive partition structures

Probabilistic divide-and-conquer: a new exact simulation method, with integer partitions as an example

We propose a new method, probabilistic divide-and-conquer, for improving the success probability in rejection sampling. For the example of integer partitions, there is an ideal recursive scheme which improves the rejection cost from asymptotically order $n^{3/4}$ to a constant. We show other examples for which a non--recursive, one--time application of probabilistic divide-and-conquer removes a substantial fraction of the rejection sampling cost. We also present a variation of probabilistic divide-and-conquer for generating i.i.d. samples that exploits features of the coupon collector's proble…

Bernoulli Sieve

Bernoulli sieve is a recursive construction of a random composition (ordered partition) of integer $n$. This composition can be induced by sampling from a random discrete distribution which has frequencies equal to the sizes of component intervals of a stick-breaking interval partition of $[0,1]$. We exploit Markov property of the composition and its renewal representation to derive asymptotics of the moments and to prove a central limit theorem for the number of parts.

正则结构中的递归公式

Recursive formulae in regularity structures

本文通过为结构群和重整化群引入递归公式，构造了正则结构的重整化模型，涵盖所有已处理的奇异SPDEs实例，并改进了基于Hopf代数的重整化程序。