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COUPLINGS, COMPONENT COUNTING PROCESSES, AND PROBABILISTIC NUMBER THEORY

  • Author(s): Squillace, Joseph Paul
  • Advisor(s): Cranston, Michael
  • Kaplan, Nathan
  • et al.
Abstract

Our results are concerned with couplings, component counts of combinatorial objects, and

probabilistic number theory. In the theory of couplings, we are concerned with the general

problem of proving the existence of joint distributions p(i,j) of two discrete random variables M and

N subject to infinitely many constraints of the form p(M=i, N=j)=0. The constraints placed on the joint distributions will require, for many

elements j in the range of N, p(M=i, N=j)=0 for infinitely many values of i in the

range of M, where the corresponding values of i depend on j. To prove the existence of

such joint distributions, we apply a theorem proved by Volker Strassen on the existence of joint

distributions with prespecified marginal distributions. In the case in which N is uniformly

distributed in a combinatorial structure with C_i components of size i, we seek to measure

the amount of dependence in the process (C_i )_{i\le n} by coupling N with a variable M such that

M has Z_i components of size i and the Z_i's are independent, with

sum_{i\le n}(C_i - Z_i)^+ <2.

In the combinatorial example of noncrossing partitions, we provide

two derivations of the probability distribution of the component counts of a uniformly distributed noncrossing partition. Upon applying a bijection between the set of noncrossing partitions

and Dyck paths consisting of up-steps and down-steps, our results specify the joint and marginal distributions of the block counts of the number of consecutive up-steps in a uniformly random chosen Dyck path.

In number theory, we give an analogue of the Erd\"{o}s-Kac

Theorem by providing a family of integer-valued random variables on

{1,\ldots,n} whose number of distinct prime factors

is roughly log\log n+X\cdot\sqrt{\log\log n} for large values

of n, where X is a standard normal variable. Our final result

involves couplings of a Zeta-distributed variable. Given s>1 and

n\in\mathbb{N}, consider a Zeta(s)-distributed integer-valued

random variable with prime factorization Z(s)=\prod_{p}p^{\alpha_{p}(s)}

and the truncation Z_{n}(s)\coloneqq\prod_{p\le n}p^{\alpha_{p}(s)

The prime powers \alpha_{p}(s) are independent with

\alpha_{p}(s)\sim\text{Geometric}(1/p^s),

and we also consider a random variable M(n)=\prod_{p\le n}p^{Z_{p}},

where the Z_{p}'s are independent with Z_{p}\sim\text{Geometric}(1/p).

We apply the concept of pivot mass and a theorem proved by Strassen

in order to prove the existence of couplings of a Zeta-distributed random variable Z(s) and M(n)

in which we can make probabilistic divisibility statements of the form "Z_{n}(s) divides M(n)P(n)"

for some random prime P(n)\le n. In particular, we will

prove that for each n\in\mathbb{N} and an integer k\ge 4, there exists an \varepsilon(k)>0

such that when s\in\left(1,1+\varepsilon(k) we can

couple Z(s) and M\left(n\right) such that if Z(s)\le k,

then Z_{n}(s) always divides M(n)P(n)

for some random prime P(n)\le n.

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