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Connective constants of lattices
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Numbers
lattice
$d$ 
$\mu$
honeycomb $(6^3)$
2:
1.847759065022573512256366378793576573644833251727284972230195462561070015002204717429679869700689192
comment: $\mu=\sqrt{2+\sqrt2}=2\cos\frac{\pi}{8}$, exact: conjectured by Nienhuis [2] and proved by Duminil-Copin and Smirnov [3]; the largest root of $x^4-4x^2+2$, OEIS A179260 [18]; $\cos\frac{\pi}{8}$ is in the table of $\cos(\pi x)$; the honeycomb lattice is also called the hexagonal lattice.
square $(4^4)$
2:
2.63815853032790 +/- 3e-14
comment: $2.63815853032790(3)$ [6], from the topological transfer matrix; the conjecture of Guttmann that $\mu$ is the positive root of $13x^4-7x^2-581$, which is $2.63815853034\ldots$ [21], fails in the twelfth digit; the first thirteen digits are OEIS A387897 [20].
triangular $(3^6)$
2:
4.150797226 +/- 26e-9
comment: $4.150797226(26)$ [7], from series for self-avoiding polygons of up to 60 steps; the triangular lattice is also called the hexagonal lattice by some authors, a name this table keeps for the honeycomb lattice.
kagome $(3,6,3,6)$
2:
2.560576765 +/- 10e-9
comment: $2.560576765(10)$ [8], from unpublished enumerations of self-avoiding polygons; the earlier $2.56062$ of Jensen and Guttmann [4] is the value listed in [16].
$(3,12^2)$
2:
1.711041296844848464117087463104454067993219326924819597700807858394925023831872933803864308022778666
comment: $\mu$ is the one real root of $x^3-\mu_{6^3}x-\mu_{6^3}$ with $\mu_{6^3}=\sqrt{2+\sqrt2}$ the honeycomb constant, and the largest real root of $x^{12}-4x^8-8x^7-4x^6+2x^4+8x^3+12x^2+8x+2$, OEIS A249776 [19]: a walk on $(3,12^2)$ is a walk on the honeycomb lattice with every visited vertex replaced by a triangle, crossed by one edge or by two [4] [5]; the lattice is also called the three-twelve or truncated hexagonal lattice.
$(4,8^2)$
2:
1.80883001 +/- 6e-8
comment: $1.80883001(6)$ [4], from series analysis, as quoted in [8] and [9]; the lattice is also called the four-eight, bathroom-tile or truncated square lattice.
Manhattan
2:
1.733535 +/- 2e-6
comment: $1.733535(2)$ [10], from series for self-avoiding polygons of up to 84 steps; the table in [16] lists the uncertainty as $3$ in the last digit.
L-lattice
2:
1.5657 +/- 15e-4 †
comment: $1.5657(15)$ as listed in [16], which names no source for this value; no paper stating it could be read for this table, and the walk counts of OEIS A322419 [22] are consistent with it.
simple cubic
3:
4.684039931 +/- 27e-9
comment: $4.684039931(27)$ [11], from the pivot algorithm; the enumeration of Schram, Barkema and Bisseling [13] gives $4.684043(12)$.
body-centred cubic
3:
6.530511501 +/- 84e-9
comment: $6.530511501(84)$ [12], from the pivot algorithm; the enumeration of Schram, Barkema, Bisseling and Clisby [14] gives $6.530520(20)$.
face-centred cubic
3:
10.03705785 +/- 14e-8
comment: $10.03705785(14)$ [12], from the pivot algorithm; the enumeration of Schram, Barkema, Bisseling and Clisby [14] gives $10.037075(20)$.
hypercubic $\mathbb{Z}^d$
4:
6.774043 +/- 5e-6
comment: $6.774043(5)$ [15], from the pivot algorithm.
hypercubic $\mathbb{Z}^d$
5:
8.838544 +/- 3e-6
comment: $8.838544(3)$ [15], from the pivot algorithm.
hypercubic $\mathbb{Z}^d$
6:
10.878094 +/- 4e-6
comment: $10.878094(4)$ [15], from the pivot algorithm.
hypercubic $\mathbb{Z}^d$
7:
12.902817 +/- 3e-6
comment: $12.902817(3)$ [15], from the pivot algorithm.
hypercubic $\mathbb{Z}^d$
8:
14.919257 +/- 2e-6
comment: $14.919257(2)$ [15], from the pivot algorithm.
Definition
For an infinite lattice graph $L$, let $c_n$ be the number of $n$-step self-avoiding walks on $L$ from a fixed vertex, following the directions of the edges if $L$ is directed, a walk being self-avoiding if it visits no vertex twice; the connective constant of $L$ [16] is $\mu(L)=\lim_{n\to\infty}c_n^{1/n}$.
Parameters
lattice
—   lattice, named by its tiling, crystal structure or vertex configuration
$d$
—   dimension ($d=2$ for the planar lattices, $d=3$ for the cubic lattices, $d\geq 4$ for the hypercubic lattice $\mathbb{Z}^d$)
Formulas
(1)
$c_{n+m}\leq c_n c_m$ for all $n,m\geq 0$, since cutting an $(n+m)$-step self-avoiding walk after $n$ steps gives an $n$-step and an $m$-step walk; hence $\mu=\lim_{n\to\infty}c_n^{1/n}=\inf_{n\geq 1}c_n^{1/n}$ exists [1] and $c_n\geq\mu^n$ for every $n\geq 1$.
(2)
$\mu(6^3)=\sqrt{2+\sqrt2}=2\cos\frac{\pi}{8}=1.8477590650\ldots$ [3], the largest root of $x^4-4x^2+2$.
(3)
$\mu(3,12^2)^3=\mu(6^3)\left(\mu(3,12^2)+1\right)$ [4], so that $\mu(3,12^2)=1.7110412968\ldots$ is the largest real root of $x^{12}-4x^8-8x^7-4x^6+2x^4+8x^3+12x^2+8x+2$, which is $(x+1)^4\left(y^4-4y^2+2\right)$ with $y=\frac{x^3}{x+1}$.
(4)
$\mu(L)\leq z-1$ for a lattice $L$ in which every vertex has degree $z$, since a self-avoiding walk never reverses a step, and $\mu\leq 2$ on the Manhattan and L lattices, where every vertex has two outgoing edges.
(5)
$\mu(\mathbb{Z}^d)=\sigma-\frac{1}{\sigma}-\frac{2}{\sigma^2}-\frac{11}{\sigma^3}-\frac{62}{\sigma^4}+O(\sigma^{-5})$ with $\sigma=2d-1$, as $d\to\infty$, as quoted in [15].
Comments
(6)
The lattices are the nearest-neighbour graphs of the tilings and crystal structures named. An Archimedean lattice is named by its vertex configuration in the notation of Grünbaum and Shephard, the cycle of polygons around a vertex, so that $(3,12^2)$ has a triangle and two dodecagons at every vertex; $(3,12^2)$ is also called the three-twelve or truncated hexagonal lattice and $(4,8^2)$ the four-eight, bathroom-tile or truncated square lattice. The honeycomb lattice $(6^3)$ is the hexagonal lattice of many authors, a name that others give to the triangular lattice $(3^6)$; this table uses honeycomb and triangular. The Manhattan lattice and the L-lattice are the square lattice with every edge directed, and a walk on them follows the directions. On the Manhattan lattice all edges of a row point the same way and consecutive rows point in opposite ways, and the same holds for the columns, like one-way streets; a walk may go straight or turn. On the L-lattice the vertices form a checkerboard of two kinds: at a vertex of the first kind both horizontal edges point in and both vertical edges point out, at a vertex of the second kind both vertical edges point in and both horizontal edges point out, so that a walk turns at every step and its consecutive steps trace the letter L. The hypercubic lattices $\mathbb{Z}^2$ and $\mathbb{Z}^3$ are the square and simple cubic rows. Every lattice here is vertex-transitive, so $c_n$ does not depend on the starting vertex.
(7)
Two of the constants are known exactly. The honeycomb constant $\sqrt{2+\sqrt2}=2\cos\frac{\pi}{8}$ was conjectured by Nienhuis [2] from the Coulomb-gas description of the $O(n)$ model at $n=0$ and proved by Duminil-Copin and Smirnov [3] with a parafermionic observable; it is the largest root of $x^4-4x^2+2$, OEIS A179260 [18], and its square $2+\sqrt2$ is in the table of algebraic numbers of degree 2. The $(3,12^2)$ constant follows from it, as Jensen and Guttmann [4] observed: replacing every vertex of the honeycomb lattice by a triangle gives $(3,12^2)$, a walk on $(3,12^2)$ crosses each triangle it visits by one edge or by two, and so a honeycomb step of weight $y$ becomes $x^2+x^3$ in the step weight $x$ of $(3,12^2)$. The two constants therefore satisfy $\frac{1}{\mu(6^3)}=\frac{1}{\mu(3,12^2)^2}+\frac{1}{\mu(3,12^2)^3}$, and $\mu(3,12^2)$ is the largest real root of the polynomial of degree $12$ in (3), OEIS A249776 [19]. Guttmann, Parviainen and Rechnitzer [5] give the full generating function, correcting an error in the count of walks by Jensen and Guttmann [4] that does not affect the constant. No connective constant of a lattice in three or more dimensions is known exactly, and no connective constant of a planar lattice other than these two.
(8)
The other constants are numerical estimates, each written as $\mu\pm\varepsilon$ with $\varepsilon$ the uncertainty stated in the paper cited in the entry, and in the notation $4.150797226(26)$ the digits in parentheses are that uncertainty in the last digits written. They are the most precise published values as of 6 September 2026, read from the papers cited: the topological transfer-matrix value of Jacobsen, Scullard and Guttmann [6] for the square lattice, the series values of Jensen [7] for the triangular lattice and [8] for the kagome lattice, of Jensen and Guttmann [4] for the $(4,8^2)$ lattice and of Bennett-Wood, Cardy, Enting, Guttmann and Owczarek [10] for the Manhattan lattice, and the pivot-algorithm values of Clisby [11] [12] and of Owczarek and Prellberg [15] for the cubic and hypercubic lattices. The L-lattice value is the one in the table in [16], which names no source for it. The square-lattice constant was long conjectured to be the positive root of $13x^4-7x^2-581$, which is $2.63815853034\ldots$, OEIS A156816 [21]; the estimate of [6] differs from it in the twelfth digit, and the first thirteen digits of the estimate are OEIS A387897 [20]. Rigorous bounds are known for every lattice here: for the planar lattices those of Jensen [8] and Alm [9], for example $2.625622<\mu(4^4)<2.679193$, and for $\mathbb{Z}^d$ those quoted by Owczarek and Prellberg [15]; every estimate lies inside its bounds.
Programs
(P1)
Sage
mu = sqrt(2 + sqrt(2))                             # honeycomb lattice, exact
N(mu, digits=30)
R.<x> = QQ[]
p = x^12 - 4*x^8 - 8*x^7 - 4*x^6 + 2*x^4 + 8*x^3 + 12*x^2 + 8*x + 2
p.roots(RealIntervalField(200))                    # (3,12^2): the largest real root
References
[1]
J. M. Hammersley, Percolation processes II. The connective constant, Proceedings of the Cambridge Philosophical Society 53 (1957), 642–645.
[2]
B. Nienhuis, Exact critical point and critical exponents of O(n) models in two dimensions, Physical Review Letters 49 (1982), 1062–1065.
[3]
H. Duminil-Copin and S. Smirnov, The connective constant of the honeycomb lattice equals $\sqrt{2+\sqrt2}$, Annals of Mathematics 175 (2012), 1653–1665. (arXiv)
[4]
I. Jensen and A. J. Guttmann, Self-avoiding walks, neighbour-avoiding walks and trails on semiregular lattices, Journal of Physics A 31 (1998), 8137–8145.
[5]
A. J. Guttmann, R. Parviainen and A. Rechnitzer, Self-avoiding walks and trails on the $3.12^2$ lattice, Journal of Physics A 38 (2005), 543–554. (arXiv)
[6]
J. L. Jacobsen, C. R. Scullard and A. J. Guttmann, On the growth constant for square-lattice self-avoiding walks, Journal of Physics A 49 (2016), 494004. (arXiv)
[7]
I. Jensen, Self-avoiding walks and polygons on the triangular lattice, Journal of Statistical Mechanics: Theory and Experiment (2004), P10008. (arXiv)
[8]
I. Jensen, Improved lower bounds on the connective constants for two-dimensional self-avoiding walks, Journal of Physics A 37 (2004), 11521–11529. (arXiv)
[9]
S. E. Alm, Upper and lower bounds for the connective constants of self-avoiding walks on the Archimedean and Laves lattices, Journal of Physics A 38 (2005), 2055–2080.
[10]
D. Bennett-Wood, J. L. Cardy, I. G. Enting, A. J. Guttmann and A. L. Owczarek, On the non-universality of a critical exponent for self-avoiding walks, Nuclear Physics B 528 (1998), 533–552. (arXiv)
[11]
N. Clisby, Calculation of the connective constant for self-avoiding walks via the pivot algorithm, Journal of Physics A 46 (2013), 245001. (arXiv)
[12]
N. Clisby, The growth constant for self-avoiding walks on the fcc and bcc lattices, Journal of Physics A 55 (2022), 465003.
[13]
R. D. Schram, G. T. Barkema and R. H. Bisseling, Exact enumeration of self-avoiding walks, Journal of Statistical Mechanics: Theory and Experiment (2011), P06019. (arXiv)
[14]
R. D. Schram, G. T. Barkema, R. H. Bisseling and N. Clisby, Exact enumeration of self-avoiding walks on BCC and FCC lattices, Journal of Statistical Mechanics: Theory and Experiment (2017), 083208. (arXiv)
[15]
A. L. Owczarek and T. Prellberg, Scaling of self-avoiding walks in high dimensions, Journal of Physics A 34 (2001), 5773–5780. (arXiv)
Links
Similar tables
Site and bond percolation thresholds of lattices —   the percolation thresholds of the same lattices
Critical exponents of the two-dimensional universality classes —   holds the exponents $\gamma=\frac{43}{32}$ and $\nu=\frac34$ of the planar self-avoiding walk, with $c_n\sim A\mu^n n^{\gamma-1}$
$\cos(\pi x)$ for rational $x$ —   holds $\cos\frac{\pi}{8}$, half the honeycomb constant
Algebraic numbers of degree 2 —   holds $2+\sqrt2$, the square of the honeycomb constant
Pólya's random walk constants —   the return probability of the simple random walk on $\mathbb{Z}^d$, the other walk constant tabulated by dimension
Data properties
Entries are of type: real number
How well the digits are known: measured
Sources of data: [3], [6], [7], [8], [4], [10], [16], [11], [12], [15]
Table is complete: no (it holds the honeycomb, square, triangular, kagome, $(3,12^2)$ and $(4,8^2)$ lattices, the Manhattan and L lattices, the simple cubic, body-centred cubic and face-centred cubic lattices and the hypercubic lattices $\mathbb{Z}^d$ for $4\leq d\leq 8$, the most precise published value of each as of 6 September 2026)
How they were obtained:

Mixed, and labelled by its weakest entries.

more

Two entries are exact and are computed in ball arithmetic in arb at 100 digits, with 64 guard bits: the honeycomb constant from $\sqrt{2+\sqrt2}$, and the $(3,12^2)$ constant as the one real root of $x^3-\mu(6^3)x-\mu(6^3)$, whose discriminant $\mu(6^3)^2(4\mu(6^3)-27)$ is negative, bracketed by bisection on the sign of the cubic with every sign decided in ball arithmetic, a sign change checked across the written enclosure, and the polynomial of degree $12$ checked to vanish on it. The other 14 entries were not computed: each is a published estimate, written as centre and radius with the paper's stated uncertainty as the radius. The L-lattice value is taken from the table in the Wikipedia article, which names no source for it, since no paper stating it could be read for this table; the Manhattan entry carries the uncertainty $2\cdot 10^{-6}$ printed in the paper rather than the $3$ in the last digit listed in that table; the $(4,8^2)$ value of Jensen and Guttmann was read as quoted by Jensen 2004 and Alm 2005, since the 1998 paper itself could not be read. Before any entry was written, the two closed forms were compared with OEIS A179260 and A249776 to every digit those entries give and with $x^4-4x^2+2$ and the polynomial of degree $12$ by a sign change across each written enclosure, and the polynomial of degree $12$ with the cubic relation by exact polynomial arithmetic; every estimate was compared with the text of the paper cited, read from its arXiv version or, for Clisby 2022, from its abstract, and with the Wikipedia table on 6 September 2026; the inequality $c_n\geq\mu^n$ was checked against the published counts $c_n$ of eleven of the lattices for every $n$ available, at most $51$; each estimate was checked to lie inside the rigorous bounds of Jensen 2004, Alm 2005 and Owczarek and Prellberg 2001; and the $1/(2d-1)$ series against the hypercubic entries.



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