The Lipatov argument

Lipatov's argument gives a formula for evaluating asymptotically the large order perturbation coefficients for the anharmonic oscillator or (φ⁴) quantum field models. We give a partial justification of the argument which enables us to prove that the radius of convergence of the Borel transform of the pressure for lattice φ⁴ models given by $$\exp \left {\mathop {\inf }\limits_\phi \left\{ {\tfrac{1}{2}\sum\limits_j {\left {(\nabla \phi )^2 (j) + \phi (j)^2 } \right] - \log } \sum {\phi (j)^4 } } \right\} - 2} \right].$$