# Electrical varieties as vertex integrable statistical models

## TL;DR

This paper introduces a novel integrable statistical model based on electrical networks, linking response matrices and boundary partition functions, and explores their mathematical structure as deformations of Lusztig varieties.

## Contribution

It develops a new vertex integrable model for electrical networks using the Local Yang-Baxter equation, connecting electrical varieties to Lusztig varieties as deformations.

## Key findings

- Response matrices can be recovered from boundary partition functions.
- Electrical varieties are shown to be deformations of Lusztig varieties.
- The approach reveals new mathematical structures on electrical varieties.

## Abstract

We propose a new approach to studying electrical networks interpreting the Ohm law as the operator which solves certain Local Yang-Baxter equation. Using this operator and the medial graph of the electrical network we define a vertex integrable statistical model and its boundary partition function. This gives an equivalent description of electrical networks We show that in the important case of an electrical network on the standard graph introduced by Curtis, Ingerman and Morrow, the response matrix of an electrical network, its most important feature, and the boundary partition function of our statistical model can be recovered from each other. Defining the electrical varieties in the usual way we compare them to the theory of the Lusztig varieties developed by Berenstein, Fomin and Zelevinsky. In our picture the former turns out to be a deformation of the latter. We describe how our approach produces new interesting mathematical structures on the electrical varieties. Our results should be compared to the earlier work started in Lam and Pylyavskyy on the connection between the Lusztig varieties and the electrical varieties.