Phase Transitions in Turnpike Theory For Mean-Field Games

We study a translation-invariant mean-field game on the flat torus with interaction $F(x,m)=\gamma (K\ast m)(x)$, 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 ${\sigma }_{\xi }(\gamma )={\nu }^2(2\pi \mathrm{\mid }\xi \mathrm{\mid }{)}^4+\gamma (2\pi \mathrm{\mid }\xi \mathrm{\mid }{)}^2\widehat{K}(\xi )$. If $\widehat{K}$ is negative for some mode, a finite threshold $${\gamma }_c=\underset{\widehat{K}(\xi )<0}{\min }\frac{{\nu }^2(2\pi \mathrm{\mid }\xi \mathrm{\mid }{)}^2}{\mathrm{\mid }\widehat{K}(\xi )\mathrm{\mid }}$$ marks loss of stability; otherwise ${\gamma }_c=+\mathrm{\infty }$. Near criticality, the spectral gap scales as $\rho (\gamma )\sim C_{\ast }\sqrt{{\gamma }_c-\gamma }$. 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 $T^{-1\mathrm{/}2}$. 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.