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Site and bond percolation thresholds of lattices
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Numbers
lattice
$d$
percolation 
$p_c$
square $(4^4)$
2
site:
0.592746050788 +/- 6e-12
comment: The published determinations disagree beyond their stated errors in the twelfth decimal: $0.59274605079210(2)$ [8] from the eigenvalue formulation of critical polynomials, $0.592746050786(3)$ [17] from the exact spanning probabilities of $n\times n$ squares with $n\leq 24$, and $0.5927460507896(1)$ [18] and $0.59274605079016(1)$ [19], the last two as quoted in [20]. The entry is the interval $0.592746050788\pm 6\cdot 10^{-12}$, which contains all four with their error bars; the first ten decimals, $0.5927460507$, are OEIS A377420 [21].
square $(4^4)$
2
bond:
1/2
comment: $p_c=\frac12$, exact: conjectured from series and self-duality by Sykes and Essam [1] and proved by Kesten [2].
equals: Rational_numbers#1/2
triangular $(3^6)$
2
site:
1/2
comment: $p_c=\frac12$, exact [1]: the triangular lattice is self-matching, so its site threshold is $\frac12$ by the argument of Kesten [2].
equals: Rational_numbers#1/2
triangular $(3^6)$
2
bond:
0.3472963553338606977034332535386295920007513543681387744724827562641316442780294708430332263147991480
comment: $p_c=2\sin\frac{\pi}{18}=2\cos\frac{4\pi}{9}$, exact, the root in $(0,1)$ of $x^3-3x+1$, OEIS A130880 [24]; found by Sykes and Essam [1] from the star–triangle transformation and proved by Wierman [3]; $\cos\frac{4\pi}{9}$ is in the table of $\cos(\pi x)$.
honeycomb $(6^3)$
2
site:
0.697040230 +/- 5e-9
comment: $0.697040230(5)$ [7], from critical polynomials on bases of up to $8$ unit cells; the honeycomb lattice is also called the hexagonal lattice.
honeycomb $(6^3)$
2
bond:
0.6527036446661393022965667464613704079992486456318612255275172437358683557219705291569667736852008520
comment: $p_c=1-2\sin\frac{\pi}{18}$, exact, the root in $(0,1)$ of $x^3-3x^2+1$, OEIS A178959 [22]; found by Sykes and Essam [1] and proved by Wierman [3]; $1$ minus the bond threshold of the triangular lattice, its planar dual.
kagome $(3,6,3,6)$
2
site:
0.6527036446661393022965667464613704079992486456318612255275172437358683557219705291569667736852008520
comment: $p_c=1-2\sin\frac{\pi}{18}=1-2\cos\frac{4\pi}{9}$, exact [1], the root in $(0,1)$ of $x^3-3x^2+1$, OEIS A178959 [22]; it equals the bond threshold of the honeycomb lattice, since the kagome lattice is the line graph of the honeycomb lattice; $\cos\frac{4\pi}{9}$ is in the table of $\cos(\pi x)$.
kagome $(3,6,3,6)$
2
bond:
0.52440499916744820 +/- 1e-17
comment: $0.52440499916744820(1)$ [9], from the eigenvalue formulation of critical polynomials; not known exactly, and the value $0.52442971\ldots$ conjectured by Wu [10] differs from it in the fifth decimal.
$(3,12^2)$
2
site:
0.8079007641202843312833520393286119147318350108627217209152260722915676700747830202460187405840713765
comment: $p_c=\sqrt{1-2\sin\frac{\pi}{18}}$, exact [6], the root in $(0,1)$ of $x^6-3x^4+1$, OEIS A174849 [23]: contracting the edges of $(3,12^2)$ that lie in no triangle gives the kagome lattice, and a contracted edge is open when both of its ends are, with probability $p^2$.
$(3,12^2)$
2
bond:
0.740420798850811610 +/- 2e-18
comment: $0.740420798850811610(2)$ [9], from the eigenvalue formulation of critical polynomials; the lattice is also called the three-twelve or truncated hexagonal lattice.
$(4,6,12)$
2
site:
0.7478008 +/- 2e-7
comment: $0.7478008(2)$ [7], from critical polynomials computed by transfer matrices; the lattice is also called the cross or truncated trihexagonal lattice.
$(4,6,12)$
2
bond:
0.693733124922 +/- 2e-12
comment: $0.693733124922(2)$ [9], from the eigenvalue formulation of critical polynomials; the lattice is also called the cross or truncated trihexagonal lattice.
$(4,8^2)$
2
site:
0.7297232 +/- 5e-7
comment: $0.7297232(5)$ [7], from critical polynomials computed by transfer matrices; the lattice is also called the four-eight, bathroom-tile or truncated square lattice.
$(4,8^2)$
2
bond:
0.6768031243900113 +/- 3e-16
comment: $0.6768031243900113(3)$ [9], from the eigenvalue formulation of critical polynomials; the lattice is also called the four-eight, bathroom-tile or truncated square lattice.
$(3,4,6,4)$
2
site:
0.62181207 +/- 7e-8
comment: $0.62181207(7)$ [7], from critical polynomials computed by transfer matrices; the lattice is also called the ruby or rhombitrihexagonal lattice.
$(3,4,6,4)$
2
bond:
0.524831461573 +/- 1e-12
comment: $0.524831461573(1)$ [9], from the eigenvalue formulation of critical polynomials; the lattice is also called the ruby or rhombitrihexagonal lattice.
$(3^4,6)$
2
site:
0.579498 +/- 3e-6 †
comment: $0.579498(3)$ [6], from hull-walk gradient percolation, with the uncertainty as listed in [20]; the paper prints $(2)$; the lattice is also called the snub hexagonal or maple-leaf lattice.
$(3^4,6)$
2
bond:
0.4343283172240 +/- 6e-13
comment: $0.4343283172240(6)$ [9], from the eigenvalue formulation of critical polynomials; the lattice is also called the snub hexagonal or maple-leaf lattice.
$(3^2,4,3,4)$
2
site:
0.550806 +/- 3e-6 †
comment: $0.550806(3)$ [6], from hull-walk gradient percolation, with the uncertainty as listed in [20]; the paper prints $(2)$; the lattice is also called the snub square, puzzle or Shastry–Sutherland lattice.
$(3^2,4,3,4)$
2
bond:
0.4141378565917 +/- 1e-13
comment: $0.4141378565917(1)$ [9], from the eigenvalue formulation of critical polynomials; the lattice is also called the snub square, puzzle or Shastry–Sutherland lattice.
$(3^3,4^2)$
2
site:
0.550213 +/- 3e-6 †
comment: $0.550213(3)$ [6], from hull-walk gradient percolation, with the uncertainty as listed in [20]; the paper prints $(2)$; the lattice is also called the frieze, trellis or elongated triangular lattice.
$(3^3,4^2)$
2
bond:
0.41964035886369 +/- 2e-14
comment: $0.41964035886369(2)$ [9], from the eigenvalue formulation of critical polynomials; the lattice is also called the frieze, trellis or elongated triangular lattice.
Cairo pentagonal $D(3^2,4,3,4)$
2
site:
0.6501834 +/- 2e-7
comment: $0.6501834(2)$ [7], from critical polynomials computed by transfer matrices.
Cairo pentagonal $D(3^2,4,3,4)$
2
bond:
0.5858621434083 +/- 1e-13
comment: $1-p_c^{\mathrm{bond}}$ of the snub square lattice $(3^2,4,3,4)$, its planar dual, whose bond threshold is $0.4141378565917(1)$ [9].
prismatic pentagonal $D(3^3,4^2)$
2
site:
0.6470471 +/- 2e-7
comment: $0.6470471(2)$ [7], from critical polynomials computed by transfer matrices.
prismatic pentagonal $D(3^3,4^2)$
2
bond:
0.58035964113631 +/- 2e-14
comment: $1-p_c^{\mathrm{bond}}$ of the elongated triangular lattice $(3^3,4^2)$, its planar dual, whose bond threshold is $0.41964035886369(2)$ [9].
floret pentagonal $D(3^4,6)$
2
site:
0.639447 +/- 5e-6 †
comment: $0.639447(5)$ [12], by simulation, the standard error of the estimate being about $5\cdot 10^{-6}$.
floret pentagonal $D(3^4,6)$
2
bond:
0.5656716827760 +/- 6e-13
comment: $1-p_c^{\mathrm{bond}}$ of the snub hexagonal lattice $(3^4,6)$, its planar dual, whose bond threshold is $0.4343283172240(6)$ [9].
rhombille $D(3,6,3,6)$
2
site:
0.585040 +/- 5e-6 †
comment: $0.585040(5)$ [12], by simulation, the standard error of the estimate being about $5\cdot 10^{-6}$; the lattice is also called the dice lattice.
rhombille $D(3,6,3,6)$
2
bond:
0.47559500083255180 +/- 1e-17
comment: $1-p_c^{\mathrm{bond}}$ of the kagome lattice $(3,6,3,6)$, its planar dual, whose bond threshold is $0.52440499916744820(1)$ [9].
deltoidal trihexagonal $D(3,4,6,4)$
2
site:
0.582410 +/- 5e-6 †
comment: $0.582410(5)$ [12], by simulation, the standard error of the estimate being about $5\cdot 10^{-6}$; the lattice is also called the ruby-dual lattice.
deltoidal trihexagonal $D(3,4,6,4)$
2
bond:
0.475168538427 +/- 1e-12
comment: $1-p_c^{\mathrm{bond}}$ of the rhombitrihexagonal lattice $(3,4,6,4)$, its planar dual, whose bond threshold is $0.524831461573(1)$ [9].
tetrakis square $D(4,8^2)$
2
site:
1/2
comment: $p_c=\frac12$, exact [1]: every face of the tetrakis square lattice is a triangle, so the lattice is self-matching.
equals: Rational_numbers#1/2
tetrakis square $D(4,8^2)$
2
bond:
0.3231968756099887 +/- 3e-16
comment: $1-p_c^{\mathrm{bond}}$ of the truncated square lattice $(4,8^2)$, its planar dual, whose bond threshold is $0.6768031243900113(3)$ [9].
kisrhombille $D(4,6,12)$
2
site:
1/2
comment: $p_c=\frac12$, exact [1]: every face of the kisrhombille lattice is a triangle, so the lattice is self-matching.
equals: Rational_numbers#1/2
kisrhombille $D(4,6,12)$
2
bond:
0.306266875078 +/- 2e-12
comment: $1-p_c^{\mathrm{bond}}$ of the truncated trihexagonal lattice $(4,6,12)$, its planar dual, whose bond threshold is $0.693733124922(2)$ [9].
triakis triangular $D(3,12^2)$
2
site:
1/2
comment: $p_c=\frac12$, exact [1]: every face of the triakis triangular lattice is a triangle, so the lattice is self-matching.
equals: Rational_numbers#1/2
triakis triangular $D(3,12^2)$
2
bond:
0.259579201149188390 +/- 2e-18
comment: $1-p_c^{\mathrm{bond}}$ of the truncated hexagonal lattice $(3,12^2)$, its planar dual, whose bond threshold is $0.740420798850811610(2)$ [9].
simple cubic
3
site:
0.31160768 +/- 15e-8
comment: $0.31160768(15)$ [13], from wrapping probabilities in Monte Carlo simulations.
simple cubic
3
bond:
0.24881185 +/- 10e-8
comment: $0.24881185(10)$ [13], from wrapping probabilities in Monte Carlo simulations; the earlier $0.24881182(10)$ of Wang, Zhou, Zhang, Garoni and Deng [14] agrees within the errors.
body-centred cubic
3
site:
0.2459615 +/- 2e-7
comment: $0.2459615(2)$ [13], from wrapping probabilities in Monte Carlo simulations.
body-centred cubic
3
bond:
0.18028762 +/- 20e-8
comment: $0.18028762(20)$ [13], from wrapping probabilities in Monte Carlo simulations.
face-centred cubic
3
site:
0.19923517 +/- 20e-8
comment: $0.19923517(20)$ [13], from wrapping probabilities in Monte Carlo simulations.
face-centred cubic
3
bond:
0.12016377 +/- 15e-8
comment: $0.12016377(15)$ [13], from wrapping probabilities in Monte Carlo simulations.
diamond
3
site:
0.4299870 +/- 4e-7
comment: $0.4299870(4)$ [13], from wrapping probabilities in Monte Carlo simulations.
diamond
3
bond:
0.3895892 +/- 5e-7
comment: $0.3895892(5)$ [13], from wrapping probabilities in Monte Carlo simulations.
hypercubic $\mathbb{Z}^d$
4
site:
0.19688561 +/- 3e-8
comment: $0.19688561(3)$ [15], from invasion percolation.
hypercubic $\mathbb{Z}^d$
4
bond:
0.16013122 +/- 6e-8
comment: $0.16013122(6)$ [15], from invasion percolation.
hypercubic $\mathbb{Z}^d$
5
site:
0.14079633 +/- 4e-8
comment: $0.14079633(4)$ [15], from invasion percolation.
hypercubic $\mathbb{Z}^d$
5
bond:
0.11817145 +/- 3e-8
comment: $0.11817145(3)$ [15], from invasion percolation.
hypercubic $\mathbb{Z}^d$
6
site:
0.109016661 +/- 8e-9
comment: $0.109016661(8)$ [15], from invasion percolation.
hypercubic $\mathbb{Z}^d$
6
bond:
0.09420165 +/- 2e-8
comment: $0.09420165(2)$ [15], from invasion percolation.
hypercubic $\mathbb{Z}^d$
7
site:
0.088951121 +/- 1e-9
comment: $0.088951121(1)$ [15], from invasion percolation.
hypercubic $\mathbb{Z}^d$
7
bond:
0.078675230 +/- 2e-9
comment: $0.078675230(2)$ [15], from invasion percolation.
hypercubic $\mathbb{Z}^d$
8
site:
0.075210128 +/- 1e-9
comment: $0.075210128(1)$ [15], from invasion percolation.
hypercubic $\mathbb{Z}^d$
8
bond:
0.0677084181 +/- 3e-10
comment: $0.0677084181(3)$ [15], from invasion percolation.
hypercubic $\mathbb{Z}^d$
9
site:
0.0652095348 +/- 6e-10
comment: $0.0652095348(6)$ [15], from invasion percolation.
hypercubic $\mathbb{Z}^d$
9
bond:
0.0594960034 +/- 1e-10
comment: $0.0594960034(1)$ [15], from invasion percolation.
hypercubic $\mathbb{Z}^d$
10
site:
0.0575929488 +/- 4e-10
comment: $0.0575929488(4)$ [15], from invasion percolation.
hypercubic $\mathbb{Z}^d$
10
bond:
0.0530925842 +/- 2e-10
comment: $0.0530925842(2)$ [15], from invasion percolation.
hypercubic $\mathbb{Z}^d$
11
site:
0.0515896843 +/- 2e-10
comment: $0.0515896843(2)$ [15], from invasion percolation.
hypercubic $\mathbb{Z}^d$
11
bond:
0.04794968373 +/- 8e-11
comment: $0.04794968373(8)$ [15], from invasion percolation.
hypercubic $\mathbb{Z}^d$
12
site:
0.0467309755 +/- 1e-10
comment: $0.0467309755(1)$ [15], from invasion percolation.
hypercubic $\mathbb{Z}^d$
12
bond:
0.04372385825 +/- 10e-11
comment: $0.04372385825(10)$ [15], from invasion percolation.
hypercubic $\mathbb{Z}^d$
13
site:
0.04271507960 +/- 10e-11
comment: $0.04271507960(10)$ [15], from invasion percolation.
hypercubic $\mathbb{Z}^d$
13
bond:
0.04018761703 +/- 6e-11
comment: $0.04018761703(6)$ [15], from invasion percolation.
Definition
For an infinite lattice graph $L$, site percolation with parameter $p$ keeps each vertex of $L$ independently with probability $p$ and bond percolation keeps each edge; the percolation threshold $p_c$ [20] is the value of $p$ above which an infinite connected component of kept vertices, or of kept edges, exists almost surely and below which it does not.
Parameters
lattice
—   lattice, named by its tiling, crystal structure or vertex configuration
$d$
—   dimension ($d=2$ for the Archimedean and Laves lattices, $d=3$ for the cubic lattices, $d\geq 4$ for the hypercubic lattice $\mathbb{Z}^d$)
percolation
—   site or bond percolation
Formulas
(1)
$p_c^{\mathrm{bond}}(4^4)=\frac12$, $p_c^{\mathrm{bond}}(3^6)=2\sin\frac{\pi}{18}$, $p_c^{\mathrm{bond}}(6^3)=p_c^{\mathrm{site}}(3,6,3,6)=1-2\sin\frac{\pi}{18}$, $p_c^{\mathrm{site}}(3^6)=\frac12$ [1] and $p_c^{\mathrm{site}}(3,12^2)=\sqrt{1-2\sin\frac{\pi}{18}}$ [6], where $2\sin\frac{\pi}{18}=2\cos\frac{4\pi}{9}=0.3472963553\ldots$ is the root in $(0,1)$ of $x^3-3x+1$ [24], $1-2\sin\frac{\pi}{18}$ the root in $(0,1)$ of $x^3-3x^2+1$ [22] and $\sqrt{1-2\sin\frac{\pi}{18}}$ the root in $(0,1)$ of $x^6-3x^4+1$ [23].
(2)
$p_c^{\mathrm{bond}}(L)+p_c^{\mathrm{bond}}(L^{*})=1$ for a planar lattice $L$ with a $2$-fold or $3$-fold rotational symmetry and its dual $L^{*}$ [4]; in particular for each Archimedean lattice and its Laves dual, and for the self-dual square lattice, where it gives $p_c^{\mathrm{bond}}(4^4)=\frac12$.
(3)
$p_c^{\mathrm{site}}(\mathcal{L}(G))=p_c^{\mathrm{bond}}(G)$ for the line graph $\mathcal{L}(G)$ of a graph $G$, since a set of edges of $G$ is a set of vertices of $\mathcal{L}(G)$ with the same connectivity; the kagome lattice is the line graph of the honeycomb lattice.
(4)
$\frac{1}{z-1}\leq p_c^{\mathrm{bond}}(L)\leq p_c^{\mathrm{site}}(L)\leq 1-\left(1-p_c^{\mathrm{bond}}(L)\right)^{z}$ for a lattice $L$ whose vertices have degree at most $z$ [5].
(5)
$p_c^{\mathrm{site}}(\mathbb{Z}^d)=\sigma^{-1}+\frac32\sigma^{-2}+\frac{15}{4}\sigma^{-3}+\frac{83}{4}\sigma^{-4}+\frac{6577}{48}\sigma^{-5}+\frac{119077}{96}\sigma^{-6}+O(\sigma^{-7})$ and $p_c^{\mathrm{bond}}(\mathbb{Z}^d)=\sigma^{-1}+\frac52\sigma^{-3}+\frac{15}{2}\sigma^{-4}+57\sigma^{-5}+\frac{4855}{12}\sigma^{-6}+O(\sigma^{-7})$ with $\sigma=2d-1$, as $d\to\infty$ [16]; the bond series through $\sigma^{-6}$ is within $10^{-6}$ of the entry for $d=13$.
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; the eight Laves lattices are the planar duals of the Archimedean lattices other than the square lattice, which is self-dual, and the triangular and honeycomb lattices, which are dual to each other, and $D(3^2,4,3,4)$ denotes the dual of $(3^2,4,3,4)$. The hypercubic lattices $\mathbb{Z}^2$ and $\mathbb{Z}^3$ are the square and simple cubic rows. Some lattices have several names: $(3,12^2)$ is the three-twelve or truncated hexagonal lattice, $(4,6,12)$ the cross or truncated trihexagonal lattice, $(4,8^2)$ the four-eight, bathroom-tile or truncated square lattice, $(3,4,6,4)$ the ruby or rhombitrihexagonal lattice, $(3^4,6)$ the snub hexagonal or maple-leaf lattice, $(3^2,4,3,4)$ the snub square, puzzle or Shastry–Sutherland lattice and $(3^3,4^2)$ the frieze, trellis or elongated triangular lattice; among the Laves lattices, $D(3,6,3,6)$ is the rhombille or dice lattice, $D(4,8^2)$ the tetrakis square or union-jack lattice, $D(4,6,12)$ the kisrhombille or bisected-hexagon lattice and $D(3,12^2)$ the triakis triangular or asanoha lattice.
(7)
Nine of the thresholds are known exactly. Sykes and Essam [1] found the bond thresholds $\frac12$ of the square lattice, $2\sin\frac{\pi}{18}$ of the triangular lattice and $1-2\sin\frac{\pi}{18}$ of the honeycomb lattice, proved by Kesten [2] for the square lattice and by Wierman [3] for the triangular and honeycomb lattices, and the site thresholds $\frac12$ of the triangular lattice and $1-2\sin\frac{\pi}{18}$ of the kagome lattice, the kagome value because the kagome lattice is the line graph of the honeycomb lattice. The site threshold $\sqrt{1-2\sin\frac{\pi}{18}}$ of $(3,12^2)$ follows from the kagome value by contracting the edges that lie in no triangle [6], and the site threshold $\frac12$ of the three Laves lattices $D(4,8^2)$, $D(4,6,12)$ and $D(3,12^2)$ holds because every face of each is a triangle, so the lattice is self-matching [1]. The bond threshold of each Laves lattice is $1$ minus the bond threshold of its Archimedean dual, by the duality theorem of Bollobás and Riordan [4]; where the dual's threshold is an estimate the Laves value inherits its uncertainty.
(8)
The other thresholds are numerical estimates, each written as $p_c\pm\varepsilon$ with $\varepsilon$ the uncertainty stated in the paper cited in the entry, usually one standard deviation, and in the notation $0.7478008(2)$ the digits in parentheses are that uncertainty in the last digits written. They are the most precise published values as of 6 September 2026, as listed in the table in [20]: for the planar lattices the critical-polynomial values of Scullard and Jacobsen [9] and Jacobsen [7], the hull-walk values of Suding and Ziff [6] and Parviainen's simulations [12]; for the cubic lattices the Monte Carlo analysis of Xu, Wang, Lv and Deng [13]; for the hypercubic lattices the invasion-percolation values of Mertens and Moore [15]. The site threshold of the square lattice is the one row whose published determinations disagree beyond their stated errors, in the twelfth decimal, and its entry is an interval holding all of them; the first ten decimals, $0.5927460507$, are OEIS A377420 [21]. The bond threshold of the kagome lattice is not known exactly: Wu [10] conjectured it to be the root in $(0,1)$ of $3p^2+6p^3-12p^4+6p^5-p^6=1$, which is $0.52442971\ldots$, and the estimate, like the earlier $0.5244053(3)$ of Ziff and Suding [11], differs from it in the fifth decimal.
Programs
(P1)
Sage
p = 2*sin(pi/18)                                   # bond threshold of the triangular lattice
N(p, digits=30), N(1 - p, digits=30), N(sqrt(1 - p), digits=30)   # and of the honeycomb lattice, and the site threshold of (3,12^2)
R.<x> = QQ[]
(x^3 - 3*x + 1).roots(RealIntervalField(200))     # 2 sin(pi/18) is the root in (0, 1)
References
[1]
M. F. Sykes and J. W. Essam, Exact critical percolation probabilities for site and bond problems in two dimensions, Journal of Mathematical Physics 5 (1964), 1117–1127.
[2]
H. Kesten, The critical probability of bond percolation on the square lattice equals 1/2, Communications in Mathematical Physics 74 (1980), 41–59.
[3]
J. C. Wierman, Bond percolation on honeycomb and triangular lattices, Advances in Applied Probability 13 (1981), 298–313.
[4]
B. Bollobás and O. Riordan, Percolation on dual lattices with k-fold symmetry, Random Structures and Algorithms 32 (2008), 463–472.
[5]
G. Grimmett, Percolation, second edition, Grundlehren der mathematischen Wissenschaften 321, Springer, 1999, chapter 1.
[6]
P. N. Suding and R. M. Ziff, Site percolation thresholds for Archimedean lattices, Physical Review E 60 (1999), 275–283. (arXiv)
[7]
J. L. Jacobsen, High-precision percolation thresholds and Potts-model critical manifolds from graph polynomials, Journal of Physics A 47 (2014), 135001. (arXiv)
[8]
J. L. Jacobsen, Critical points of Potts and O(N) models from eigenvalue identities in periodic Temperley–Lieb algebras, Journal of Physics A 48 (2015), 454003. (arXiv)
[9]
C. R. Scullard and J. L. Jacobsen, Bond percolation thresholds on Archimedean lattices from critical polynomial roots, Physical Review Research 2 (2020), 012050. (arXiv)
[10]
F. Y. Wu, Critical point of planar Potts models, Journal of Physics C 12 (1979), L645–L650.
[11]
R. M. Ziff and P. N. Suding, Determination of the bond percolation threshold for the kagome lattice, Journal of Physics A 30 (1997), 5351–5359. (arXiv)
[12]
R. Parviainen, Connectivity Properties of Archimedean and Laves Lattices, Uppsala Dissertations in Mathematics 34, Uppsala University, 2005, Table 4.3.
[13]
X. Xu, J. Wang, J.-P. Lv and Y. Deng, Simultaneous analysis of three-dimensional percolation models, Frontiers of Physics 9 (2014), 113–119. (arXiv)
[14]
J. Wang, Z. Zhou, W. Zhang, T. M. Garoni and Y. Deng, Bond and site percolation in three dimensions, Physical Review E 87 (2013), 052107. (arXiv)
[15]
S. Mertens and C. Moore, Percolation thresholds and Fisher exponents in hypercubic lattices, Physical Review E 98 (2018), 022120. (arXiv)
[16]
S. Mertens and C. Moore, Series expansion of the percolation threshold on hypercubic lattices, Journal of Physics A 51 (2018), 475001. (arXiv)
[17]
S. Mertens, Exact site-percolation probability on the square lattice, Journal of Physics A 55 (2022), 334002. (arXiv)
[18]
Y. Yang and S. Zhou, Comment on ‘Critical points of Potts and O(N) models from eigenvalue identities in periodic Temperley–Lieb algebras’, Journal of Physics A 57 (2024), 258001.
[19]
J. L. Jacobsen, Reply to Comment on ‘Critical points of Potts and O(N) models from eigenvalue identities in periodic Temperley–Lieb algebras’, Journal of Physics A 57 (2024), 258002.
Links
Similar tables
$\cos(\pi x)$ for rational $x$ —   holds $\cos\frac{4\pi}{9}=\sin\frac{\pi}{18}$, half the bond threshold of the triangular lattice
Rational numbers —   holds $\frac12$, the threshold of five entries here, which link it
Pólya's random walk constants —   the return probability of the simple random walk on $\mathbb{Z}^d$, the other lattice probability tabulated by dimension
Data properties
Entries are of type: real number
How well the digits are known: measured
Sources of data: [20]
Table is complete: no (it holds site and bond percolation on the eleven Archimedean lattices, the eight Laves lattices, the simple cubic, body-centred cubic, face-centred cubic and diamond lattices and the hypercubic lattices $\mathbb{Z}^d$ for $4\leq d\leq 13$, the most precise published value of each as of 6 September 2026)
How they were obtained:

Mixed, and labelled by its weakest entries.

more

Nine entries are exact and are computed in ball arithmetic in arb at 100 digits from their closed forms, with 64 guard bits: $\frac12$, written as the rational, for the site thresholds of the triangular lattice and of $D(4,8^2)$, $D(4,6,12)$ and $D(3,12^2)$ and the bond threshold of the square lattice; $2\sin\frac{\pi}{18}$ for the bond threshold of the triangular lattice; $1-2\sin\frac{\pi}{18}$ for the bond threshold of the honeycomb lattice and the site threshold of the kagome lattice; and $\sqrt{1-2\sin\frac{\pi}{18}}$ for the site threshold of $(3,12^2)$. The other 57 entries were not computed: each is a published estimate, written as centre and radius with the paper's stated uncertainty as the radius, and the eight Laves bond thresholds are $1$ minus the value of the Archimedean dual, in exact decimal arithmetic with the same radius. The three site thresholds taken from Suding and Ziff carry the uncertainty $3\cdot 10^{-6}$ listed for them in the Wikipedia table rather than the $2\cdot 10^{-6}$ printed in the paper, because the values that paper gives for the lattices since measured more precisely lie two to three of its standard errors from the later values. Before any entry was written, the three closed forms were compared with OEIS A130880, A178959 and A174849 to every digit those entries give and with their minimal polynomials by a sign change across each written enclosure; every estimate was compared with the table of the paper cited, read from its arXiv version, and with the Wikipedia table on 6 September 2026; the inequalities $\frac{1}{z-1}\leq p_c^{\mathrm{bond}}\leq p_c^{\mathrm{site}}\leq 1-(1-p_c^{\mathrm{bond}})^z$ were checked on every lattice with $z$ its largest vertex degree, and the $1/(2d-1)$ series against the hypercubic entries. The square-lattice site threshold is the one entry whose interval was chosen here, to contain four published determinations that disagree beyond their stated errors; the values from the 2024 comment and reply were taken from the Wikipedia table and not read from those papers.



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