T8-Core — Phase Transition (Spectral Universality)

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1. #1

   Let be a finite connected graph with Fiedler eigenvalue (algebraic connectivity). Let with volume fraction

   $$c\in \left(\frac{3-\sqrt{3}}{6},\frac{3+\sqrt{3}}{6}\right)\approx (0.211,0.789),$$
2. #2

   and let be the double-well potential. If

   $$\boxed{\frac{\beta }{\alpha }>\frac{4{\lambda }_2}{|W^{\prime \prime }(c)|}}$$
3. #3

   The second variation of at along the Fiedler eigenvector (eigenvalue ) has eigenvalue

   $$4\alpha {\lambda }_2+\beta W^{\prime \prime }(c).$$
4. #4

   The double-well derivative satisfies on the spinodal interval . Therefore, when

   $$4\alpha {\lambda }_2+\beta W^{\prime \prime }(c)<0⟺\beta /\alpha >4{\lambda }_2/|W^{\prime \prime }(c)|,$$