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Explore the tropicalization of scalar quantum field theory in this advanced physics lecture from the Institute for Advanced Study's Amplitudes Group Meeting. Learn how tropicalized massive scalar quantum field theory becomes exactly solvable through a non-linear recursion equation that governs the expansion coefficients of the quantum effective action. Discover the geometric interpretation of this recursion as it computes specific volumes of moduli spaces of metric graphs, drawing parallels to Mirzakhani's volume recursions on the moduli space of curves. Examine the construction of a polynomial-time algorithm that samples points from the moduli space of graphs proportional to their perturbative contribution, suggesting that perturbative quantum field theory computations may belong to the polynomial-time complexity class despite existing exponential-time algorithms for individual Feynman integrals. Review practical demonstrations including the evaluation of the primitive contribution to the phi4 beta function at 50 loops and phi3 theory correlation functions in D=3 up to 20 loops using a proof-of-concept implementation.
