An upper bound for the connective constant of the square lattice

We prove that the connective constant μ of the square lattice Z², the exponential growth rate of the number of n-step self-avoiding walks, satisfies μ ≤ 2.6586513. The previous best upper bound, μ ≤ 2.662342426, is due to Couronné. Like the bounds of Pönitz and Tittmann and of Couronné, ours comes from a finite automaton that remembers a bounded part of the past of the walk. The new idea is to count bridges. Since they can all be preceded by the same infinite history, the automaton may treat as visited the unvisited cells that no infinite self-avoiding continuation can reach, and three tests on the current state and the next step recognize such cells. The bound is checked in exact integer arithmetic and proved in the Lean proof assistant.

The programs and the Lean formalization are archived at doi:10.5281/zenodo.23715897, and the three files of the certificate, compressed and split into pieces, in five records: doi:10.5281/zenodo.23574056, doi:10.5281/zenodo.23574058, doi:10.5281/zenodo.23574060, doi:10.5281/zenodo.23574062 and doi:10.5281/zenodo.23574064. The programs and the Lean files are also available at github.com/tejasvi/connective-constant-bound/tree/v8.