The Forgetfulness of Balls and Bins

TL;DR

This paper analyzes the asymptotic total variation distance between two distributions of ball placements in labeled bins, comparing uniform random placement with a planted configuration plus random placement.

Contribution

It introduces a precise asymptotic analysis of the total variation distance between these two distributions, highlighting the effect of a fixed planted arrangement.

Findings

Quantifies the total variation distance asymptotically

Shows the impact of planted configurations on distribution similarity

Provides tools for analyzing planted versus uniform distributions

Abstract

We find the asymptotic total variation distance between two distributions on configurations of m balls in n labeled bins: in the first, each ball is placed in a bin uniformly at random; in the second, k balls are planted in an arbitrary but fixed arrangement and the remaining m-k balls placed uniformly at random.