# Locality of interactions for planar memristive circuits

## TL;DR

This paper investigates the non-local effects in planar memristive circuits, providing exact results on how interactions decay with circuit topology, which is crucial for understanding their behavior in applications like on-chip machine learning.

## Contribution

It offers a formalism that captures Kirchhoff constraints as a projection operator and derives exact results on interaction decay with Hamming distance in memristive circuits.

## Key findings

- Interaction strength decreases with Hamming distance
- Exact decay rates for memristive circuit elements
- Clarifies non-local effects in crossbar arrays

## Abstract

Memristors are nonlinear passive circuit elements which can be thought as time varying resistances. When connected in a complex circuit these exhibit very exotic behavior, typical of disordered systems, such as a universal slow relaxation for intricated circuit topologies, and strong dependence on the initial conditions. Being memristive components part of a circuit, non-local effects due to the Kirchhoff constraints are present. In the formalism developed recently for a fairly general class of memristive circuits the constraints are contained in a projection operator. We provide exact results regarding the fall-off of the elements with the Hamming distance on the circuit, thus elucidating an insofar elusive and open question regarding the non-local effects in crossbar arrays, currently being considered for on-chip machine learning.