Square-lattice self-avoiding walk connective constant $\mu_{\mathbb{Z}^2}$

Description of constant

Let $\mathbb{Z}^2$ denote the square lattice graph with vertex set $\mathbb{Z}^2$ and edges between nearest neighbors (Euclidean distance $1$).

A self-avoiding walk (SAW) on a graph $G=(V,E)$ is a walk that visits no vertex more than once. In particular, for $\ell=1,2,\dots$ and $v\in V$, let $N(v,\ell)$ denote the number of SAWs in $G$ of length $\ell$ starting at $v$. [SSSY2014-Nv-ell]

The connective constant (also called the SAW growth constant) of a graph $G$ is defined by

\(\mu(G)\ :=\ \sup_{v\in V}\ \limsup_{\ell\to\infty} N(v,\ell)^{1/\ell}.\) [SSSY2014-def-cc]

For vertex-transitive graphs, the $\limsup$ in the definition above can be replaced by a true limit. [SSSY2014-rem-vtx-limit]

For the square lattice $G=\mathbb{Z}^2$, let $c_n$ be the number of $n$-step SAWs starting at the origin. Then the square-lattice SAW connective constant is

Known upper bounds

Known lower bounds

Additional comments and links

• Status (rigorous bounds vs. numerical estimate). The exact value of $\mu_{\mathbb{Z}^2}$ is not known. [SlaBounds-unknown] Moreover, [FV2017] reports the rigorous interval \(2.625622\ <\ \mu_{\mathbb{Z}^2}\ <\ 2.679193,\) and also records a high-precision numerical estimate $\mu_{\mathbb{Z}^2}\approx 2.63815853032790(3)$. [FV2017-bounds-square] [FV2017-estimate-square] [FV2017-ref-180]

• Historical origin of the notion. A modern reference notes that “the first reference to the connective constant” appears in classical work of Hammersley and collaborators (1950s). [SSSY2014-history]

• Surveys/background: [FV2017], [SSSY2014].

References

• [FV2017] Friedli, Roland; Velenik, Yvan. Statistical Mechanics of Lattice Systems: a Concrete Mathematical Introduction. Cambridge University Press (2017). DOI: 10.1017/9781316882603. Google Scholar. Author PDF
  • [FV2017-bounds-square]
    loc: Unige PDF p.149 (discussion of the square-lattice connective constant).
    quote: “The more precise bounds on the connective constant $2.625622 < \mu < 2.679193$ can be found in [182] and [277] respectively.”
  • [FV2017-ref-182]
    loc: Unige PDF p.553, References [182].
    quote: “[182] Iwan Jensen. Improved lower bounds on the connective constants for two-dimensional self-avoiding walks. J. Phys. A, 37(48):11521–11529, 2004.”
  • [FV2017-ref-277]
    loc: Unige PDF p.557, References [277].
    quote: “[277] André Pönitz and Peter Tittmann. Improved upper bounds for self-avoiding walks in Zd . Electron. J. Combin., 7:Research Paper 21, 10 pp. (electronic), 2000.”
  • [FV2017-estimate-square]
    loc: Unige PDF p.149 (discussion of the square-lattice connective constant).
    quote: “Numerically, the best estimate at the moment of writing seems to be $\mu \simeq 2.63815853032790(3)$ [180].”
  • [FV2017-ref-180]
    loc: Unige PDF p.553, References [180].
    quote: “[180] Jesper Lykke Jacobsen, Christian R. Scullard, and Anthony J. Guttmann. On the growth constant for square-lattice self-avoiding walks. J. Phys. A, 49(49):494004, 18, 2016.”
• [SSSY2014] Sinclair, Alistair; Srivastava, Piyush; Štefankovič, Daniel; Yin, Yitong. Spatial mixing and the connective constant: Optimal bounds. Probability Theory and Related Fields 168 (2017), 153–197. DOI: 10.1007/s00440-016-0708-2. Google Scholar. arXiv PDF.
  • [SSSY2014-Nv-ell]
    loc: arXiv PDF p.3, Section 1.2 (Contributions), paragraph introducing the connective constant.
    quote: “Given a graph $G$ and a vertex $v$ in $G$, let $N (v, \ell)$ denote the number of self avoiding walks in $G$ of length $\ell$ starting at $v$.”
  • [SSSY2014-def-cc]
    loc: arXiv PDF, Section 2.5, Definition 2.6.
    quote: “Definition 2.6 (Connective constant: infinite graphs [30]). Let $G = (V, E)$ be a locally finite infinite graph. The connective constant $\Delta(G)$ of $G$ is $\sup_{v \in V} \limsup_{\ell \to \infty} N(v, \ell)^{1/\ell}$.”
  • [SSSY2014-rem-vtx-limit]
    loc: arXiv PDF, Section 2.5, Remark 2.4 (vertex-transitive graphs).
    quote: “Further, in such graphs the lim sup can be replaced by a limit [30].”
  • [SSSY2014-history]
    loc: arXiv PDF, Section 1 (Introduction).
    quote: “The first reference to the connective constant occurs in the classical papers by Hammersley and Morton [18], Hammersley and Broadbent [8] and Hammersley [17].”
• [SlaBounds] Slade, Gordon. Bounds on the self-avoiding-walk connective constant. In: Benedetto, John J. (ed.), The Journal of Fourier Analysis and Applications. CRC Press (2020), 525–533. DOI: 10.1201/9780429332838-32. Google Scholar. Author PDF.
  • [SlaBounds-unknown]
    loc: PDF p.2 (Introduction, paragraph on unknown precise value).
    quote: “The precise value of $\mu$ is of course not known in any dimension $d \ge 2$.”
  • [SlaBounds-simple]
    loc: PDF p.2 (Introduction, paragraph on simplest bounds).
    quote: “The simplest bounds on $\mu$ are $d \le \mu \le 2d - 1$.”
  • [SlaBounds-table1-d2]
    loc: PDF p.3, Table 1 (row $d=2$).
    quote: “d lower bound estimate upper bound 2 2.620 02a 2.638 158 5 (10)d 2.695 76b”.
  • [SlaBounds-conway-guttmann]
    loc: PDF p.2 (Introduction, paragraph on best bounds for $d=2$).
    quote: “For d = 2 the best lower bound is due to Conway and Guttmann [3] and makes use of extensive walk enumerations; it also is described below.”

• [Alm1993] Alm, Sven Erick. Upper bounds for the connective constant of self-avoiding walks. Combinatorics, Probability and Computing 2(2) (1993), 115–136. Google Scholar. Publisher entry.

• [Jen2004] Jensen, Iwan. Enumeration of self-avoiding walks on the square lattice. Journal of Physics A: Mathematical and General 37(21) (2004), 5503–5524. Google Scholar. Publisher entry.

• [CG1993] Conway, A. R.; Guttmann, A. J. Lower bound on the connective constant for square lattice self-avoiding walks. Journal of Physics A: Mathematical and General 26 (1993), 3719–3724. Google Scholar. Publisher entry.

• [Jen2004-lb] Jensen, Iwan. Improved lower bounds on the connective constants for two-dimensional self-avoiding walks. Journal of Physics A: Mathematical and General 37(48) (2004), 11521–11529. Google Scholar. Publisher entry.

• [PT2000] Pönitz, André; Tittmann, Peter. Improved upper bounds for self-avoiding walks in $\mathbb{Z}^d$. Electronic Journal of Combinatorics 7 (2000), R21. DOI: 10.37236/1499. Google Scholar. EJC PDF.

• [JSG2016] Jacobsen, Jesper Lykke; Scullard, Christian R.; Guttmann, Anthony J. On the growth constant for square-lattice self-avoiding walks. Journal of Physics A: Mathematical and Theoretical 49(49) (2016), 494004. Google Scholar. Publisher entry.

• [HM1954] Hammersley, J. M.; Morton, K. W. Poor man’s Monte Carlo. Journal of the Royal Statistical Society. Series B (Methodological) 16 (1954), 23–38. Google Scholar. Publisher entry.

• [HB1957] Hammersley, J. M.; Broadbent, S. R. Percolation processes I. Crystals and mazes. Proceedings of the Cambridge Philosophical Society 53(3) (1957), 629–641. Google Scholar. Publisher entry.

• [Ham1957] Hammersley, J. M. Percolation processes: lower bounds for the critical probability. The Annals of Mathematical Statistics 28(3) (1957), 790–795. Google Scholar. Publisher entry.

• [MS1996] Madras, Neal; Slade, Gordon. The Self-Avoiding Walk. Birkhäuser (1996). Google Scholar. Publisher entry.

Contribution notes

Prepared with assistance from ChatGPT 5.2 Pro.