Critical-line conjecture for the disordered two-community noisy Kuramoto model

Let $\mu$ be the common distribution of natural frequencies in the two communities, let $\psi=\psi_{2}-\psi_{1}\in\{0,\pi\}$, and let $r_{1},r_{2}$ be the two synchronization order parameters. Define

$$\chi=\int_{\mathbb{R}}\mu(\mathrm{d}\omega)\,\frac{1}{2(1+4\omega^{2})}.$$

Assume that $\mu$ is symmetric and unimodal. Critical-line conjecture. The parameter space $(K,L)\in\mathbb{R}^{2}$ has the following two regimes:

• If $K+L\cos\psi\leq\chi^{-1}$, the only solution is the unsynchronized solution $(r_{1},r_{2})=(0,0)$.
• If $K+L\cos\psi>\chi^{-1}$, there are at least two solutions: $(r_{1},r_{2})=(0,0)$ and a symmetric synchronized solution $(r_{1},r_{2})=(r,r)$ for some $r\in(0,1)$.

This identifies the expected synchronization threshold when the two communities have identical symmetric unimodal disorder and the phase difference is $0$ or $\pi$. The surrounding discussion derives the critical line heuristically; excluding solutions with one order parameter zero is described as non-trivial.