OP-12048OpenProbability

Square-lattice self-avoiding walk connective constant ( $C_{38}$ )

Square-lattice self-avoiding walk connective constant: determine $C_{38}$ or improve its bounds; the repository records 2.6273856 (lower) and 2.679193 (upper).

MSC 82B41 · Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics MSC 60K35 · Interacting random processes; statistical mechanics type models; percolation theory

Editorial MSC2020 classification. Original article codes and source evidence are recorded in the literature directory.

Licensed source statementAs of Repository table at commit 2c1968c (2026-09-26); open while the recorded lower and upper bounds differSource & evidence ↓

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Problem statement

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

C_{38} := \mu_{\mathbb{Z}^2}\ :=\ \lim_{n\to\infty} c_n^{1/n}.

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Background

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 (from the repository page)

[Num2026] Numaro (numaro.tech). An improved rigorous lower bound for the square-lattice SAW connective constant. Certificate archive, submitted to this repository (2026).
[Num2026-bound] loc: certificate archive, check_saw38a_L17.py output and README. quote: "certified: mu(Z^2) >= R = 2.627385640... (exact dyadic 340282366920938463463374607431768211456/129513673858566230428673965530967449326); ALL CERTIFIED."
[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.

Progress & source assessment

Repository table (commit 2c1968c, 26 September 2026): best known lower bound 2.6273856; best known upper bound 2.679193. Values marked with an asterisk have minimal verification.

Known upper bounds

Bound	Reference	Comments
$3$	Trivial	From the general bound $d \le \mu \le 2d-1$ with $d=2$ . [SlaBounds-simple]
$2.69576$	[SlaBounds]	Reported (Table 1) as the best rigorous upper bound for $d=2$ in this survey, attributed there to [Alm1993]. [SlaBounds-table1-d2]
$2.679193$	[FV2017]	Reported as a rigorous upper bound in [FV2017] (attributed there to [PT2000]). [FV2017-bounds-square] [FV2017-ref-277]

Known lower bounds

Bound	Reference	Comments
$2$	Trivial	From the general bound $d \le \mu \le 2d-1$ with $d=2$ . [SlaBounds-simple]
$2.62002$	[SlaBounds]	Reported (Table 1) as the best rigorous lower bound for $d=2$ in this survey; the survey attributes it to [CG1993]. [SlaBounds-table1-d2] [SlaBounds-conway-guttmann]
$2.625622$	[FV2017]	Reported as a rigorous lower bound in [FV2017] (attributed there to [Jen2004-lb]). [FV2017-bounds-square] [FV2017-ref-182]
$2.6273856$	[Num2026]	Kesten irreducible-bridge renewal exactly as in [Jen2004-lb], extended two span levels past the 2004 computation (span $L=17$ , series length $N=260$ ) by an exact cell-by-cell finite-lattice transfer matrix (CRT big-integer arithmetic); the certificate is a single exact big-integer inequality (self-contained checker included), and secondary certificates at Jensen's own span ( $L=15$ , $N=260$ gives $2.6256270$ ) and at $L=16$ ( $2.6265705$ ) independently also exceed $2.625622$ . [Num2026-bound]

References

[1]Davis, Damek; Ivanisvili, Paata; Tao, Terence (maintainers). Optimization Constants in Mathematics, github.com/teorth/optimizationproblems (commit 2c1968c, 26 September 2026). — Square-lattice self-avoiding walk connective constant ($C_{38}$) linkApache-2.0 (repository license); constant pages reproduced with attribution; repository references retained.

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Source & evidence

https://teorth.github.io/optimizationproblems/constants/38a.html

Constant page from github.com/teorth/optimizationproblems at commit 2c1968c, Apache-2.0. The maintainers review submissions for plausibility and replicability but do not certify correctness; asterisked bounds have minimal verification.

Source date: Repository table at commit 2c1968c (2026-09-26); open while the recorded lower and upper bounds differ
Imported: 2026-10-10

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Difficulty: HardResisted sustained effort by specialists.
Posed by: Davis, Damek; Ivanisvili, Paata; Tao, Terence (maintainers) and contributors
Posed: 2026
Source: Curated problem listOptimization Constants in Mathematics — Davis, Ivanisvili & TaoOptimization Constants in Mathematics — Davis, Ivanisvili & TaoOriginal statement ↗
Citation: Davis, Damek; Ivanisvili, Paata; Tao, Terence (maintainers). Optimization Constants in Mathematics, github.com/teorth/optimizationproblems (commit 2c1968c, 26 September 2026). — Square-lattice self-avoiding walk connective constant ($C_{38}$)
Added: 5 Nov 2026
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Probability