Uniqueness of synchronized stationary equilibria in the Kuramoto mean field game

TL;DR

This paper proves the uniqueness and smooth bifurcation structure of synchronized Nash equilibria in the Kuramoto mean field game, confirming a conjecture about the scalar self-consistency map.

Contribution

It establishes the strict concavity of the self-consistency map and characterizes the bifurcation of synchronized equilibria, settling a prior conjecture.

Findings

The synchronized branch is a unique smooth family of Nash equilibria.

The scalar self-consistency map is strictly concave.

Synchronized equilibria converge smoothly to the uniform distribution at the critical threshold.

Abstract

The stationary Kuramoto mean field game models a population of phase oscillators that form synchronized Nash equilibria above a critical interaction strength. We prove that the synchronized branch is a unique smooth family of Nash equilibria emerging from the uniform state at the bifurcation: at each supercritical interaction strength the synchronized equilibrium is unique up to rotation of the torus, and converges smoothly to the uniform distribution as the interaction parameter decreases to the critical threshold. Both follow from our main technical result: the scalar self-consistency map is strictly concave, settling a conjecture of Carmona, Cormier, and Soner. The proof decomposes the second derivative of the self-consistency map into two sign-indefinite moments of the equilibrium--a cubic moment and a gradient moment--and controls their signs through sharp shape estimates for the value function, a pointwise geometric-mean monotonicity that determines the sign of the cubic moment via a cosine-skewness inequality, and a reflection argument combined with a correlation inequality for the gradient moment.