TejasviTomar
October 2026 · Preprint

An upper bound for the connective constant of the square lattice

Tejasvi Singh Tomar

In plain words

Self-avoiding walks on a square grid never visit a point twice. Their count grows roughly like a fixed number, the connective constant, raised to the number of steps. Its exact value is unknown; estimates put it near 2.638. This paper lowers Couronné's upper bound of 2.662342426 to 2.6600756. It follows bridges, walks that can be extended forever and grow in number at the same rate, keeping a finite memory of part of their past and of points they could enter only by trapping themselves. Two separately written programs check the proof's table of 4,124,965,633 memories and weights.

Abstract

Let μ be the connective constant of self-avoiding walks on the square lattice Z2, the exponential growth rate of the number of n-step self-avoiding walks. We prove μ ≤ 2.6600756. The previous best upper bound, μ ≤ 2.662342426, is due to Couronné. As in the work of Pönitz and Tittmann and of Couronné, the bound comes from a finite automaton whose states remember a bounded part of the past of the walk. Our automaton counts bridges, and, as in Couronné’s automaton, its states may contain cells that the walk has not visited; here these are cells that no infinite self-avoiding continuation of the walk can visit, recognized by three tests that depend only on the state and the step. Distinct bridges, preceded by a fixed infinite history, then give distinct paths in the automaton, so the Hammersley–Welsh theorem turns a bound for the number of paths into a bound for μ. The bound follows from a certificate, consisting of the automaton (4,124,965,633 states) and a vector of integer weights, which a program checks in exact integer arithmetic.

Forty small teal lattice paths in five rows of eight, on white with a faint grid of gray dots and no axes or labels. Each path is a chain of short horizontal and vertical segments with a small magenta dot at one end and a larger yellow dot at the other, which the path always reaches from the left. The first path, at the top left, is a straight horizontal line; the others bend into staircases, hooks and curls of different shapes.
Illustration of the certificate: 40 of its 4,124,965,633 states, taken at equal spacing through its sorted list of keys (every 103,124,100th key, from the first), one per tile at a common scale on a faint grid of lattice points. A state is a set of cells, written as a lattice path that ends at the current position, the yellow dot, which the path enters from the left; the magenta dot is the first cell of the path. Every cell of a state is either visited or dead, that is, no infinite self-avoiding continuation of the current past can visit it. The keys sort the states by their number of cells: the first tile is the start state, a straight path of 15 cells, and the shown states have 15 to 29 cells. When the walk takes a step, the automaton passes to a state made of cells of the old state, the new position and new dead cells, and the largest ratio of the summed weights of a state's children, one per step, to its own weight, 7653257817410405/2877082878637577 = 2.660075548826…, is the bound. The states are decoded from the published key file as the paper's checker decodes them.