# Displacement Convexity in Spatially Coupled Scalar Recursions

## TL;DR

This paper introduces a displacement convexity framework for analyzing spatially coupled scalar recursions, revealing conditions for unique fixed points and broad applicability across coding, sensing, and statistical models.

## Contribution

It establishes displacement convexity of the potential functional for spatially coupled systems, linking fixed points to minimizers and providing conditions for their uniqueness.

## Key findings

- Displacement convexity holds for a wide class of spatially coupled recursions.
- Fixed points correspond to potential minimizers under mild conditions.
- The framework applies to coding, compressive sensing, and statistical mechanics models.

## Abstract

We introduce a technique for the analysis of general spatially coupled systems that are governed by scalar recursions. Such systems can be expressed in variational form in terms of a potential functional. We show, under mild conditions, that the potential functional is \emph{displacement convex} and that the minimizers are given by the fixed points of the recursions. Furthermore, we give the conditions on the system such that the minimizing fixed point is unique up to translation along the spatial direction. The condition matches those in \cite{KRU12} for the existence of spatial fixed points. \emph{Displacement convexity} applies to a wide range of spatially coupled recursions appearing in coding theory, compressive sensing, random constraint satisfaction problems, as well as statistical mechanical models. We illustrate it with applications to Low-Density Parity-Check and generalized LDPC codes used for transmission on the binary erasure channel, or general binary memoryless symmetric channels within the Gaussian reciprocal channel approximation, as well as compressive sensing.