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

   $$\boxed{\;\frac{\beta}{\alpha} \;>\; \frac{4\lambda_2}{|W''(c)|}\;}$$
3. #3

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

   $$4\alpha\lambda_2 + \beta W''(c).$$
4. #4

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

   $$4\alpha\lambda_2 + \beta W''(c) < 0 \quad\Longleftrightarrow\quad \beta/\alpha > 4\lambda_2/|W''(c)|,$$