# Concavity analysis of the tangent method

## TL;DR

This paper provides a universal geometric framework explaining the tangent method's effectiveness in determining arctic curves, highlighting the role of entropy concavity in lattice path models.

## Contribution

It introduces a general geometric explanation for the tangent method's tangency property based on entropy concavity, extending the proof to q-deformed paths.

## Key findings

- Entropy concavity underpins the tangent method's tangency property.
- The framework applies to models with directed lattice paths.
- Extension of proof to q-deformed lattice paths.

## Abstract

The tangent method has recently been devised by Colomo and Sportiello (arXiv:1605.01388 [math-ph]) as an efficient way to determine the shape of arctic curves. Largely conjectural, it has been tested successfully in a variety of models. However no proof and no general geometric insight have been given so far, either to show its validity or to allow for an understanding of why the method actually works. In this paper, we propose a universal framework which accounts for the tangency part of the tangent method, whenever a formulation in terms of directed lattice paths is available. Our analysis shows that the key factor responsible for the tangency property is the concavity of the entropy (also called the Lagrangean function) of long random lattice paths. We extend the proof of the tangency to $q$-deformed paths.