Prize Talk: Dannie Heineman Prize for Mathematical PhysicsTitle: The Lace expansion and Random Walk Representation in Statistical Mechanics

For a large class of classical ferromagnetic lattice spin systems that includes the Ising model, the φ^4 lattice euclidean field theory and O(N) models, the two point function has a Random Walk Representation as a sum over self-interacting walks. This a prolific source of correlation inequalities. Applications include Froehlich's 1982 proof that continuum limits of φ^4 are Gaussian in five or more dimensions and a short construction of continuum φ^4 in two and three dimensions. The random walk representation shows that self-avoiding walk is "part" of φ^4. A graphical expansion for self-avoiding walk is obtained by expanding the self-interaction, but has far too many graphs to be absolutely convergent. However, there exists a resummation that simultaneously removes infrared divergences and eliminates most of the graphs. The resulting Lace Expansion is convergent for self-avoiding walk in five or more dimensions, and convergence implies that the end-to-end distance of the walk grows as the square root of the number of steps. Similar lace resummations are possible for many systems: in 1990 Hara and Slade derived a lace expansion for critical percolation which converges in high dimensions and proves that the critical exponents of percolation have mean field values. Recently the random walk expansion was used to derive a lace expansion for critical φ^4 which converges in five dimensions if the coupling constant is small and convergence implies that the two point function of critical lattice φ^4 equals the two point function of the massless free field with a correction that decays more rapidly.

The φ^4 lace expansion does not converge in the critical dimension four because it does not renormalise the coupling constant. Is there a convergent resummation that includes coupling constant renormalisation? Can lace expansions prove rotational invariance of continuum limits of lattice models?