# The entropic regularization of the Monge problem on the real line

## TL;DR

This paper investigates the entropic regularization of the one-dimensional optimal transport problem with a distance cost, identifying the limiting plan as the most diffuse among optimal solutions.

## Contribution

It characterizes the limit of entropically regularized optimal transport plans in one dimension with the distance cost, highlighting the plan's diffuseness.

## Key findings

- The limiting plan is the most diffuse among optimal plans.
- Entropic regularization selects a unique, diffuse solution in the limit.
- The results provide insights into the structure of optimal transport plans under regularization.

## Abstract

We study the entropic regularization of the optimal transport problem in dimension 1 when the cost function is the distance c(x, y) = |y -- x|. The selected plan at the limit is, among those which are optimal for the non-penalized problem, the most 'diffuse' one on the zones where it may have a density.