The connective constant μ of an infinite, locally finite graph G (most commonly a regular lattice such as ℤ²) is the exponential growth rate of the number cₙ of n‑step self‑avoiding walks (SAWs) starting from a fixed origin vertex. Formally
$$ \mu(G)=\lim_{n\to\infty} c_{n}^{1/n}, $$
where the limit exists by subadditivity (Fekete’s lemma). The quantity μ is independent of the chosen origin when G is transitive.
Key properties
- Existence: For any infinite, connected, locally finite graph the limit defining μ exists (Hammersley, 1957).
- Monotonicity: If H is a subgraph of G that contains the origin, then μ(H) ≤ μ(G).
- Bounds: Simple counting arguments give 1 ≤ μ ≤ Δ − 1, where Δ is the maximum degree of G. More refined bounds are known for specific lattices.
Values for common lattices
| Lattice (dimension) | Approximate μ | Remarks |
|---|---|---|
| Hexagonal (honeycomb, 2‑D) | √(2 + √2) ≈ 1.847759 | Proven rigorously (Duminil‑Copin & Smirnov, 2012). |
| Square (ℤ²) | ≈ 2.638158530 | Numerical estimates from series expansions and Monte‑Carlo simulations. |
| Triangular (2‑D) | ≈ 4.150 | Numerical estimates; no closed form known. |
| Simple cubic (ℤ³) | ≈ 4.684 | Obtained via computational methods. |
Applications
- Polymer physics: The connective constant governs the entropy of long polymer chains modeled as SAWs.
- Statistical mechanics: It appears in the study of the lattice animal problem and percolation theory.
- Combinatorics: Provides asymptotic information for enumerative problems involving SAWs and related objects.
References
- Hammersley, J. M. (1957). “Percolation processes: Lower bounds for the critical probability.” Proceedings of the Cambridge Philosophical Society.
- Madras, N., & Slade, G. (1993). The Self‑Avoiding Walk. Birkhäuser.
- Duminil‑Copin, H., & Smirnov, S. (2012). “The connective constant of the honeycomb lattice.” Annals of Mathematics, 175(3), 1653‑1665.
