Phase Transitions in Turnpike Theory For Mean-Field Games

We study a translation-invariant mean-field game on the flat torus with interaction $F(x,m)=\gamma (K*m)(x)$, where $K$ is smooth, even, and mean-zero. The interaction is of potential type, arising as the first variation of a quadratic energy, though the stationary system is not treated variationally. Linearizing around the uniform equilibrium yields mode-wise $2\times 2$ systems with dispersion $\sigma_\xi(\gamma)=\nu^2(2\pi|\xi|)^4+\gamma(2\pi|\xi|)^2\hat K(\xi)$. If $\hat K$ is negative for some mode, a finite threshold $$\gamma_c=\min_{\hat K(\xi)<0}\frac{\nu^2(2\pi|\xi|)^2}{|\hat K(\xi)|}$$ marks loss of stability; otherwise $\gamma_c=+\infty$. Near criticality, the spectral gap scales as $\rho(\gamma)\sim C_*\sqrt{\gamma_c-\gamma}$. For $\gamma<\gamma_c$, the uniform state is exponentially stable in the turnpike sense for finite-horizon problems, with rate $\rho(\gamma)$. At $\gamma=\gamma_c$, the gap closes and, after phase fixing and center-manifold reduction, one obtains algebraic midpoint decay of order $T^{-1/2}$. For $\gamma>\gamma_c$, a branch of nonuniform stationary solutions bifurcates via a pitchfork-type amplitude equation, with translations generating the full family. Finally, under standard asymptotic-consistency assumptions on symmetric $N$-player equilibria in the subcritical regime, we obtain qualitative propagation of chaos, without quantitative rates.