Entropy constants of lattice models

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Numbers

model

lattice

expression

$\kappa$ or $h$

hard-core gas (independent sets)

line $\mathbb{Z}$

$\kappa$:

1.618033988749894848204586834365638117720309179805762862135448622705260462818902449707207204189391137

comment: $\kappa=\varphi=\frac{1+\sqrt5}{2}$, the golden ratio: the path on $n$ vertices has $F_{n+2}$ independent sets, and the one-dimensional hard-core gas is the golden-mean shift.
equals: Golden_ratio#phi

hard-core gas (independent sets)

line $\mathbb{Z}$

$h=\ln\kappa$:

0.4812118250596034474977589134243684231351843343856605196610181688401638676082217744120094291227234750

comment: $h=\ln\varphi$, the topological entropy of the golden-mean shift and the regulator of $\mathbb{Q}(\sqrt5)$.
equals: Regulators_of_real_quadratic_fields#5

hard-core gas (independent sets)

square $(4^4)$

$\kappa$:

1.5030480824753322643220663294755536893857810

comment: The hard-square entropy constant, OEIS A085850 [19]: not known in closed form; the $43$ decimals are Baxter's corner-transfer-matrix value [3], and the first $25$ are proved by the bounds $1.50304808247533226432206632947<\kappa<1.50304808247533226432206633030$ of Liang [5].

hard-core gas (independent sets)

square $(4^4)$

$h=\ln\kappa$:

0.40749510126068800045014681235865045422368

comment: $h=\ln\kappa$ from Baxter's $43$ decimals [3], OEIS A379041 [20]; the topological entropy of the two-dimensional golden-mean shift, the standard $\mathbb{Z}^2$ shift of finite type whose entropy has no known closed form.

hard-core gas (independent sets)

triangular $(3^6)$

$\kappa$:

1.395485972479302735229500663566888068954103728144661190817472156135760880358697774689837873085275428

comment: The hard-hexagon entropy constant, OEIS A085851 [21], from Baxter's exact solution [2] evaluated at activity $z=1$; an algebraic number of degree $24$ [4], the root of the polynomial (10). Baxter gives $55$ decimals [3]; the last digit listed in A085851 is one too small.

hard-core gas (independent sets)

triangular $(3^6)$

$h=\ln\kappa$:

0.3332427219761818878537477640056763435594703357229638837842080301598018906370972505328961681792752254

comment: $h=\ln\kappa$ from the exact solution; Metcalf and Yang conjectured $h=\frac13$, which Baxter and Tsang refuted before Baxter solved the model [3].

hard-core gas (independent sets)

honeycomb $(6^3)$

$\kappa$:

1.54644070878756141848902270530472278

comment: Baxter's corner-transfer-matrix value [3], printed there to $38$ decimals of which "the last two or three digits should be treated with caution", so $35$ are kept.

hard-core gas (independent sets)

honeycomb $(6^3)$

$h=\ln\kappa$:

0.4359559734410476815991032706617904

comment: $h=\ln\kappa$ from the $35$ decimals kept of Baxter's value [3].

ice model (Eulerian orientations)

square $(4^4)$

$\kappa$:

1.539600717839002038691063414671886548393604670053671669382939537290607126141155588516574388228665401

comment: $\kappa=\left(\frac43\right)^{3/2}=\frac{8\sqrt3}{9}$, Lieb's square ice constant [6], OEIS A118273 [22]; the growth rate of the proper $3$-colourings of the square lattice.

ice model (Eulerian orientations)

square $(4^4)$

$h=\ln\kappa$:

0.4315231086776713911588285089907411472552645663466415847599985280239394260811706965071663487686579294

comment: $h=\frac32\ln\frac43$, the residual entropy of square ice per vertex in units of Boltzmann's constant [6].

dimer model (perfect matchings)

square $(4^4)$

$\kappa$:

1.338515151976096766938195902018513537064353697127911314641234786622391133007980978646487384617744539

comment: $\kappa=e^{G/\pi}$ with $G$ Catalan's constant [7] [9], OEIS A097469 [23]; the number of domino tilings per cell of the board, that is per vertex of the lattice. Per dimer the constant is $\kappa^2=e^{2G/\pi}=1.7916228\ldots$, OEIS A130834 [25].

dimer model (perfect matchings)

square $(4^4)$

$h=\ln\kappa$:

0.2915609040308187801383844564683949188640661539858372702610015691117476368804388617266268243031340589

comment: $h=G/\pi$ [7], OEIS A143233 [24]; $G=L(2,\chi_{-4})$ is in the table of Dirichlet $L$-values.

dimer model (perfect matchings)

triangular $(3^6)$

$\kappa$:

1.535098483272734137069289006038665026948796856053784183294104972609604690818919684539242687550327784

comment: $\kappa=e^{h}$ with $h$ the Kasteleyn integral (5) of Fendley, Moessner and Sondhi [10]; per dimer the constant is $\kappa^2=2.3565273\ldots$, OEIS A247548 [26].

dimer model (perfect matchings)

triangular $(3^6)$

$h=\ln\kappa$:

0.4285945374649588653584255564520155133111245974494630836695954222555248414097766638980597225348640087

comment: $h=\frac{1}{16\pi^2}\int_{-\pi}^{\pi}\!\int_{-\pi}^{\pi}\ln\left(6+2\cos u+2\cos v+2\cos(u+v)\right)\,\mathrm{d}u\,\mathrm{d}v$ [10], who give $0.4286$.

dimer model (perfect matchings)

honeycomb $(6^3)$

$\kappa$:

1.175311211772651215599813186791236659885532007387707742635421848799592912335156217526580001540879280

comment: $\kappa=e^{h}$; the number of lozenge tilings per vertex of the honeycomb lattice [8].

dimer model (perfect matchings)

honeycomb $(6^3)$

$h=\ln\kappa$:

0.1615329736097252570468182553619031970361209203902935080654342351805075564036349210418938045446856960

comment: $h=\frac{3\sqrt3}{8\pi}L(2,\chi_{-3})=\frac12 m(1+x+y)$, half the Mahler measure of $1+x+y$ [11], and one tenth of the spanning-tree constant of the triangular lattice; $L(2,\chi_{-3})$ is in the table of Dirichlet $L$-values.

spanning trees

square $(4^4)$

$\kappa$:

3.209912300728157678629749481779905158748592124251834494874586005846102464162424020406676712151410887

comment: $\kappa=e^{4G/\pi}$, OEIS A229728 [28]: spanning trees per vertex of the square lattice.

spanning trees

square $(4^4)$

$h=\ln\kappa$:

1.166243616123275120553537825873579675456264615943349081044006276446990547521755446906507297212536236

comment: $z_{\mathrm{sq}}=\frac{4G}{\pi}=\frac4\pi\left(1-\frac1{3^2}+\frac1{5^2}-\cdots\right)$ [12] [13], OEIS A218387 [27]; the Mahler measure of $4+x+x^{-1}+y+y^{-1}$ [14].

spanning trees

triangular $(3^6)$

$\kappa$:

5.029546072970906422186745870406623973669638516279899198827902912273509836816879007458803733878244240

comment: $\kappa=e^{z_{\mathrm{tri}}}$.

spanning trees

triangular $(3^6)$

$h=\ln\kappa$:

1.615329736097252570468182553619031970361209203902935080654342351805075564036349210418938045446856960

comment: $z_{\mathrm{tri}}=\frac{3\sqrt3}{\pi}\left(1-\frac1{5^2}+\frac1{7^2}-\frac1{11^2}+\frac1{13^2}-\cdots\right)=\frac{15\sqrt3}{4\pi}L(2,\chi_{-3})$ [12] [13], OEIS A245725 [29].

spanning trees

honeycomb $(6^3)$

$\kappa$:

2.242664948888020237028773931912807542795631222873299251361962513256207274076752068017801783089205771

comment: $\kappa=e^{z_{\mathrm{hc}}}$.

spanning trees

honeycomb $(6^3)$

$h=\ln\kappa$:

0.8076648680486262852340912768095159851806046019514675403271711759025377820181746052094690227234284802

comment: $z_{\mathrm{hc}}=\frac12 z_{\mathrm{tri}}$ [13], by the duality (9), OEIS A245737 [30].

spanning trees

kagome $(3,6,3,6)$

$\kappa$:

3.113340924279582590607529456410547982636362178684166191368668056189257750714216803035170364118389361

comment: $\kappa=e^{z_{\mathrm{kag}}}$.

spanning trees

kagome $(3,6,3,6)$

$h=\ln\kappa$:

1.135696401775102523760219970666578081028066632028646595503238898311987826408217630966139042419002579

comment: $z_{\mathrm{kag}}=\frac13\left(z_{\mathrm{tri}}+\ln 6\right)$ [13], OEIS A245739 [31].

spanning trees

diced $D(3,6,3,6)$

$\kappa$:

3.113340924279582590607529456410547982636362178684166191368668056189257750714216803035170364118389361

comment: $\kappa=e^{z_{\mathrm{kag}}}$: the diced lattice is the planar dual of the kagome lattice and has the same vertex density, so the two share their constant [13].

spanning trees

diced $D(3,6,3,6)$

$h=\ln\kappa$:

1.135696401775102523760219970666578081028066632028646595503238898311987826408217630966139042419002579

comment: $z_{\mathrm{diced}}=z_{\mathrm{kag}}$ [13], by the duality (9).

spanning trees

$(3,12^2)$

$\kappa$:

2.055590845879885707113290176216496540146858958934287667467234650725810280795254675322108282364948614

comment: $\kappa=e^{z}$ with $z=\frac16\left(z_{\mathrm{tri}}+\ln 15\right)$ [13].

spanning trees

$(3,12^2)$

$h=\ln\kappa$:

0.7205633228665771060773645206279575524223835193323670423836140961527914741604359903204479463922947767

comment: $z=\frac16\left(z_{\mathrm{tri}}+\ln 15\right)$ [13], who give $0.7205633$.

spanning trees

$(4,8^2)$

$\kappa$:

2.196102669202442160697401841020442464994198552685921934118164015223723179250658218268867822440267179

comment: $\kappa=e^{z}$ with $z$ the integral (7) of Shrock and Wu [13].

spanning trees

$(4,8^2)$

$h=\ln\kappa$:

0.7866842753788321791216579894946953805511708165780327497186464518988179928818399372439686672615234781

comment: $z=\frac14\ln2+\frac1{4\pi}\int_0^{\pi}\ln\left(7-3\cos\theta+4\sin\frac\theta2\sqrt{5-\cos\theta}\right)d\theta$ [13], who give $0.786684(1)$.

spanning trees

union jack $D(4,8^2)$

$\kappa$:

4.822866933678091099892364997288847432524076303535982947877790130594101785596674152160008146458081748

comment: $\kappa=e^{z}$ with $z$ twice the constant of $(4,8^2)$, its planar dual [13].

spanning trees

union jack $D(4,8^2)$

$h=\ln\kappa$:

1.573368550757664358243315978989390761102341633156065499437292903797635985763679874487937334523046956

comment: $z_{\mathrm{UJ}}=2z_{(4,8^2)}$ [13], by the duality (9); they give $1.573368(2)$.

spanning trees

simple cubic

$\kappa$:

5.330202889205167421134597996649659520108

comment: $\kappa=e^{z_{\mathrm{sc}}}$, spanning trees per vertex of the simple cubic lattice.

spanning trees

simple cubic

$h=\ln\kappa$:

1.673389302970196732283430621655598075258

comment: $z_{\mathrm{sc}}=\frac1{\pi^2}\int_0^{\pi}\!\int_0^{\pi}\operatorname{arccosh}(3-\cos\theta_1-\cos\theta_2)\,\mathrm{d}\theta_1\,\mathrm{d}\theta_2$ (8); the value $1.6741481(1)$ printed by Shrock and Wu [13] differs from it in the fourth decimal, and the closed-walk series $\ln 6-\sum_{m\geq1}W_{2m}/(2m\cdot 36^m)$ gives $1.6733893029701967322834\ldots$.

Definition

For a lattice model on an infinite lattice $L$ whose configurations on a finite region $\Lambda$ of $L$ with $|\Lambda|$ vertices are counted by $Z_\Lambda$, the entropy constant is $\kappa=\lim_{\Lambda\to L}Z_\Lambda^{1/|\Lambda|}$, the number of configurations per site, and the entropy per site is $h=\ln\kappa$ [1].

Parameters

model

— lattice model, by what $Z_\Lambda$ counts

lattice

— lattice, named by its tiling, crystal structure or vertex configuration

expression

— the constant $\kappa$ or the entropy $h=\ln\kappa$

Formulas

(1)

$\kappa(\text{hard-core},\mathbb{Z})=\varphi=\frac{1+\sqrt5}{2}$, since the path on $n$ vertices has $F_{n+2}$ independent sets, $F_n$ the Fibonacci numbers.

(2)

$\kappa(\text{ice},4^4)=\left(\frac43\right)^{3/2}=\frac{8\sqrt3}{9}$ [6].

(3)

$h(\text{dimer},4^4)=\frac{G}{\pi}=\frac1\pi\sum_{k\geq0}\frac{(-1)^k}{(2k+1)^2}=\frac{L(2,\chi_{-4})}{\pi}$ [7] [9], the number of domino tilings of an $m\times n$ rectangle being $\prod_{j=1}^{\lceil m/2\rceil}\prod_{k=1}^{\lceil n/2\rceil}\left(4\cos^2\frac{\pi j}{m+1}+4\cos^2\frac{\pi k}{n+1}\right)$.

(4)

$h(\text{dimer},6^3)=\frac{1}{16\pi^2}\int_{-\pi}^{\pi}\!\int_{-\pi}^{\pi}\ln\left(3+2\cos u+2\cos v+2\cos(u+v)\right)\,\mathrm{d}u\,\mathrm{d}v=\frac{3\sqrt3}{8\pi}L(2,\chi_{-3})$ [8], half the Mahler measure of $1+x+y$ [11], and $h(\text{dimer},6^3)=\frac1{10}z_{\mathrm{tri}}$.

(5)

$h(\text{dimer},3^6)=\frac{1}{16\pi^2}\int_{-\pi}^{\pi}\!\int_{-\pi}^{\pi}\ln\left(6+2\cos u+2\cos v+2\cos(u+v)\right)\,\mathrm{d}u\,\mathrm{d}v=\frac{1}{8\pi}\int_{-\pi}^{\pi}\ln\frac{A+\sqrt{A^2-B^2}}{2}\,\mathrm{d}u$ with $A=6+2\cos u$ and $B=4\cos\frac u2$ [10].

(6)

$z_{\mathrm{sq}}=\frac{4G}{\pi}$, $z_{\mathrm{tri}}=\frac{3\sqrt3}{\pi}\left(1-\frac1{5^2}+\frac1{7^2}-\frac1{11^2}+\frac1{13^2}-\cdots\right)=\frac{15\sqrt3}{4\pi}L(2,\chi_{-3})$, $z_{\mathrm{hc}}=\frac12z_{\mathrm{tri}}$ [12], $z_{\mathrm{kag}}=z_{\mathrm{diced}}=\frac13\left(z_{\mathrm{tri}}+\ln6\right)$ and $z_{(3,12^2)}=\frac16\left(z_{\mathrm{tri}}+\ln15\right)$ [13], where $z_L=h(\text{spanning trees},L)$ and the sum runs over the $n$ coprime to $6$ with the sign $\chi_{-3}(n)$.

(7)

$z_{(4,8^2)}=\frac14\ln2+\frac1{4\pi}\int_0^{\pi}\ln\left(7-3\cos\theta+4\sin\frac\theta2\sqrt{5-\cos\theta}\right)d\theta$ and $z_{\mathrm{UJ}}=2z_{(4,8^2)}$ for the union-jack lattice [13].

(8)

$z_{\mathrm{sc}}=\ln6+\frac{1}{(2\pi)^3}\int_{[-\pi,\pi]^3}\ln\left(1-\frac{\cos\theta_1+\cos\theta_2+\cos\theta_3}{3}\right)d^3\theta=\frac1{\pi^2}\int_0^{\pi}\!\int_0^{\pi}\operatorname{arccosh}\left(3-\cos\theta_1-\cos\theta_2\right)d\theta_1\,\mathrm{d}\theta_2=\ln6-\sum_{m\geq1}\frac{W_{2m}}{2m\cdot36^m}$ [13], with $W_{2m}=\binom{2m}{m}\sum_{k=0}^{m}\binom mk^2\binom{2k}{k}$ the number of closed walks of length $2m$ on $\mathbb{Z}^3$ [32].

(9)

$z_{L^{*}}=z_L/\nu_L$ for a planar lattice $L$ and its dual $L^{*}$, where $\nu_L$ is the number of vertices of $L^{*}$ per vertex of $L$ [13]: $\nu=\frac12$ for the honeycomb lattice and for $(4,8^2)$, and $\nu=1$ for the kagome lattice.

(10)

$\kappa(\text{hard-core},3^6)$ is a root of $11^{10}z^{24}+2013290651222784z^{22}+2505062311720673792z^{20}+797726698866658379776z^{18}+7449488310131083100160z^{16}+2958015038376958230528z^{14}-72405670285649161617408z^{12}+107155448150443388043264z^{10}-71220809441400405884928z^{8}-73347491183630103871488z^{6}+97143135277377575190528z^{4}-32751691810479015985152$ [4] [18], its only positive real root; in Baxter's parametrisation [2] it is the value at the $x\in(-1,0)$ with $-xH(x)^5/G(x)^5=1$ of $\kappa=\frac{H(x)^3Q(x^5)^2}{G(x)^2}\prod_{n\geq1}\frac{(1-x^{6n-4})(1-x^{6n-3})^2(1-x^{6n-2})}{(1-x^{6n-5})(1-x^{6n-1})(1-x^{6n})^2}$, with $G(x)=\prod_{n\geq1}\frac{1}{(1-x^{5n-4})(1-x^{5n-1})}$ and $H(x)=\prod_{n\geq1}\frac{1}{(1-x^{5n-3})(1-x^{5n-2})}$ the Rogers–Ramanujan products and $Q(x)=\prod_{n\geq1}(1-x^n)$.

Comments

(11)

The hard-core lattice gas at activity $1$ counts the independent sets of $\Lambda$, the sets of vertices no two of which are adjacent; on the square lattice it is the hard-square model [19], on the triangular lattice the hard-hexagon model [15], and on the line it is the golden-mean shift, the sequences of $0$ and $1$ with no two adjacent $1$. The ice model counts the Eulerian orientations of the square lattice, the orientations of its edges with two edges into and two out of every vertex, which are the states of the six-vertex model with all weights $1$ [16]; the same count is, up to a bounded factor, the number of proper $3$-colourings of the square lattice. The dimer model counts the perfect matchings, the domino tilings of the square lattice [17] and the lozenge tilings of the honeycomb lattice. The spanning-tree constant $z_L=\lim|\Lambda|^{-1}\ln N_{\mathrm{ST}}(\Lambda)$ of Shrock and Wu [13] is the entropy $h$ of the spanning-tree model, $N_{\mathrm{ST}}(\Lambda)$ being the number of spanning trees of $\Lambda$, and their notation $z_{\mathrm{sq}}$, $z_{\mathrm{tri}}$, $z_{\mathrm{hc}}$, $z_{\mathrm{kag}}$ is used in the entry comments.

(12)

Every constant is per site, that is per vertex of the lattice, and $h$ is the natural logarithm of $\kappa$. Finch's dimer constant [1] and OEIS A130834 [25] are per dimer, which is $\kappa^2$ here, and the square-ice constant is sometimes given per vertex of the dual lattice, which is the same number on the square lattice; the entry comments say where a source normalises differently. The limit is taken along tori, or along rectangles with free boundaries, which give the same constant; for the dimer model $|\Lambda|$ is even. The lattices are named as in the table of percolation thresholds: an Archimedean lattice by its vertex configuration, and the diced lattice $D(3,6,3,6)$ and the union-jack lattice $D(4,8^2)$ as the planar duals of the kagome lattice and of $(4,8^2)$.

(13)

Ten of the seventeen constants are known in closed form: the golden ratio for the line, Lieb's $\left(\frac43\right)^{3/2}$ for square ice [6], $e^{G/\pi}$ for dimers on the square lattice with $G$ Catalan's constant [7], the honeycomb dimer constant and the spanning-tree constants of the square, triangular, honeycomb, kagome, diced and $(3,12^2)$ lattices, which are exponentials of rational multiples of $G/\pi$ and of $\sqrt3L(2,\chi_{-3})/\pi$, times $6^{1/3}$ for the kagome and diced lattices and $15^{1/6}$ for $(3,12^2)$ [12] [13]. The hard-hexagon constant is algebraic of degree $24$ [4], from Baxter's exact solution [2]. The dimer constant of the triangular lattice and the spanning-tree constants of $(4,8^2)$, the union-jack lattice and the simple cubic lattice are integrals with no known closed form. The hard-square and honeycomb hard-core constants are not known exactly; the entries are Baxter's corner-transfer-matrix values [3], and for hard squares the first $25$ decimals are proved by the bounds of Liang [5]. The hard-core models have a phase transition at activity $z_c=\frac{11+5\sqrt5}{2}=\varphi^5$ on the triangular lattice [2], at $z_c\approx 3.7962$ on the square lattice and at $z_c\approx 7.92$ on the honeycomb lattice [3], all far larger than the activity $1$ of these entries.

Programs

(P1)

Sage

N(exp(catalan/pi), digits=30)                        # dimers on the square lattice, per site
N((4/3)^(3/2), digits=30)                            # Lieb's square ice constant
L3 = (hurwitz_zeta(2, 1/3) - hurwitz_zeta(2, 2/3))/9  # L(2, chi_{-3})
N(15*sqrt(3)/(4*pi)*L3, digits=30)                   # spanning trees on the triangular lattice, z_tri

References

[1]

S. R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications 94, Cambridge University Press, 2003, sections 5.12, 5.22, 5.23 and 5.24.

[2]

R. J. Baxter, Hard hexagons: exact solution, Journal of Physics A 13 (1980), L61–L70.

[3]

R. J. Baxter, Planar lattice gases with nearest-neighbour exclusion, Annals of Combinatorics 3 (1999), 191–203. (arXiv)

[4]

G. S. Joyce, On the hard-hexagon model and the theory of modular functions, Philosophical Transactions of the Royal Society of London A 325 (1988), 643–702.

[5]

K. Liang, Independent set enumeration and estimation of related constants of grid graphs (2025), equation (27). (arXiv)

[6]

E. H. Lieb, Residual entropy of square ice, Physical Review 162 (1967), 162–172.

[7]

P. W. Kasteleyn, The statistics of dimers on a lattice I. The number of dimer arrangements on a quadratic lattice, Physica 27 (1961), 1209–1225.

[8]

P. W. Kasteleyn, Dimer statistics and phase transitions, Journal of Mathematical Physics 4 (1963), 287–293.

[9]

H. N. V. Temperley and M. E. Fisher, Dimer problem in statistical mechanics – an exact result, Philosophical Magazine 6 (1961), 1061–1063.

[10]

P. Fendley, R. Moessner and S. L. Sondhi, Classical dimers on the triangular lattice, Physical Review B 66 (2002), 214513. (arXiv)

[11]

C. J. Smyth, On measures of polynomials in several variables, Bulletin of the Australian Mathematical Society 23 (1981), 49–63.

[12]

F. Y. Wu, Number of spanning trees on a lattice, Journal of Physics A 10 (1977), L113–L115.

[13]

R. Shrock and F. Y. Wu, Spanning trees on graphs and lattices in d dimensions, Journal of Physics A 33 (2000), 3881–3902. (arXiv)

[14]

A. J. Guttmann, Spanning tree generating functions and Mahler measure, Journal of Physics A 45 (2012), 494010. (arXiv)

Links

[15]

Wikipedia: Hard hexagon model

[16]

Wikipedia: Lieb's square ice constant

[17]

Wikipedia: Domino tiling

[18]

MathWorld: Hard Hexagon Entropy Constant

[19]

OEIS A085850: hard square entropy constant

[20]

OEIS A379041: logarithm of the hard square entropy constant

[21]

OEIS A085851: hard hexagon entropy constant

[22]

OEIS A118273: $(4/3)^{3/2}$, Lieb's square ice constant

[23]

OEIS A097469: growth constant $e^{G/\pi}$ of the dimer model on the square lattice

[24]

OEIS A143233: the dimer constant $G/\pi$

[25]

OEIS A130834: $e^{2G/\pi}$, the dimer constant of the square lattice per dimer

[26]

OEIS A247548: $D^2$, the dimer constant of the triangular lattice per dimer

[27]

OEIS A218387: spanning tree constant $4G/\pi$ of the square lattice

[28]

OEIS A229728: $e^{4G/\pi}$

[29]

OEIS A245725: $z_{\mathrm{tri}}$, spanning trees on the triangular lattice

[30]

OEIS A245737: $z_{\mathrm{hc}}$, spanning trees on the honeycomb lattice

[31]

OEIS A245739: $z_{\mathrm{kag}}$, spanning trees on the kagome lattice

[32]

OEIS A002896: closed walks of length $2n$ on the cubic lattice

Similar tables

Values of Dirichlet $L$-functions at positive integers — holds Catalan's constant $G=L(2,\chi_{-4})$ and $L(2,\chi_{-3})$, of which the closed-form entropies are rational multiples times $\frac1\pi$ or $\frac{\sqrt3}{\pi}$

Golden ratio — $\varphi$ is the constant of the hard-core gas on the line, which links it

Regulators of real quadratic fields — $\ln\varphi$ is the regulator of $\mathbb{Q}(\sqrt5)$, which the entropy of the hard-core gas on the line links

Site and bond percolation thresholds of lattices — the same lattices, under the same names, and the other family of constants of lattice statistical mechanics

Pólya's random walk constants — the return probability of the simple random walk on $\mathbb{Z}^d$; the closed walks $W_{2m}$ of the series for $z_{\mathrm{sc}}$ are the walks it counts

Feigenbaum constants — the other constants of dynamical systems here; $h$ is the topological entropy of the shift of finite type the hard-core, ice and dimer models define

Data properties

Entries are of type: real number

How well the digits are known: heuristic

Sources of data: [3], [13]

Table is complete: no (it holds $\kappa$ and $h$ for the hard-core gas on the line and on the square, triangular and honeycomb lattices, the ice model on the square lattice, the dimer model on the square, triangular and honeycomb lattices, and spanning trees on the square, triangular, honeycomb, kagome, diced, $(3,12^2)$, $(4,8^2)$, union-jack and simple cubic lattices)

How they were obtained:

Mixed, and labelled by its weakest entries.

Twenty-eight entries are enclosures computed in ball arithmetic in arb at 100 digits with 64 guard bits: the closed forms, with Catalan's constant and $L(2,\chi_{-3})$ taken from arb's Hurwitz zeta function; the two one-variable integrals for the triangular dimer constant and for $z_{(4,8^2)}$, integrated with arb's rigorous integrator after the other angle was integrated in closed form; and the hard-hexagon constant, computed from Baxter's exact solution with 220 guard bits, the root $x$ of $-xH(x)^5/G(x)^5=1$ enclosed by bisection to a width of $10^{-120}$ and every infinite product truncated with an explicit bound on its tail. Two entries, $z_{\mathrm{sc}}$ and its exponential, are heuristic: the double integral was evaluated by nested tanh-sinh quadrature in mpmath at 40 and 55 decimal digits, and the digits the two agree on are written. Four entries are transcribed from Baxter [3]: the hard-square constant with the 43 decimals he states are correct, of which Liang's bounds [5] prove 25, and the honeycomb constant with 35 of his 38 decimals, since he says the last two or three should be treated with caution; their logarithms are computed from those intervals and carry the same precision. Before any entry was written, every closed form was compared with the OEIS entry that holds it to every digit listed there; the hard-hexagon enclosure with Baxter's 55 decimals and his density $\rho(1)$, and with Joyce's polynomial as given by MathWorld, which vanishes on it; the triangular dimer integral with OEIS A247548 to 103 digits; the $(4,8^2)$ integral with Shrock and Wu's $0.786684(1)$, the same reduction giving $\frac{4G}{\pi}$ on the square lattice; and $z_{\mathrm{sc}}$ with the closed-walk series summed to $1.6\cdot 10^6$ terms and extrapolated, which agrees with the quadrature to 22 digits, so that the value $1.6741481(1)$ printed by Shrock and Wu is wrong in the fourth decimal.