Asymptotic Behavior of the Zhang--Kawazumi's phi-invariant

The phi-invariant was introduced by Shou-Wu Zhang and Nariya Kawazumi. We study the continuity property of the phi-invariants for degenerating graphs, and show that this continuity property induces an adelic divisor on the moduli space of Riemann surfaces, and then give an asymptotic expression of the Zhang–Kawazumi’s invariants for Riemann surfaces near the boundary of the moduli space. We mainly use Yuan–Zhang’s adelic divisors, and follow Yuan’s idea of globalization of phi-invariants.