History of Entropy constants of lattice models

 Display properties:   number-header: $\kappa$ or $h$-Numbers: []+Numbers:+- params:+    model: hard-core+    lattice: line+    expression: kappa+  number: '1.618033988749894848204586834365638117720309179805762862135448622705260462818902449707207204189391137'+  comment: '$\kappa=\varphi=\frac{1+\sqrt5}{2}$, the HREF{Golden_ratio}[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: HREF{Golden_ratio#phi}+- params:+    model: hard-core+    lattice: line+    expression: entropy+  number: '0.4812118250596034474977589134243684231351843343856605196610181688401638676082217744120094291227234750'+  comment: $h=\ln\varphi$, the topological entropy of the golden-mean shift and the+    HREF{Regulators_of_real_quadratic_fields#5}[regulator of $\mathbb{Q}(\sqrt5)$].+  equals: HREF{Regulators_of_real_quadratic_fields#5}+- params:+    model: hard-core+    lattice: square+    expression: kappa+  number: '1.5030480824753322643220663294755536893857810'+  comment: 'The hard-square entropy constant, OEIS A085850 CITE{OEIShs}: not known+    in closed form; the $43$ decimals are Baxter''s corner-transfer-matrix value CITE{Baxter99},+    and the first $27$ are proved by the bounds $1.50304808247533226432206632947<\kappa<1.50304808247533226432206633030$+    of Liang CITE{Liang}.'+- params:+    model: hard-core+    lattice: square+    expression: entropy+  number: '0.40749510126068800045014681235865045422368'+  comment: $h=\ln\kappa$ from Baxter's $43$ decimals CITE{Baxter99}, OEIS A379041+    CITE{OEIShsh}; 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.+- params:+    model: hard-core+    lattice: triangular+    expression: kappa+  number: '1.395485972479302735229500663566888068954103728144661190817472156135760880358697774689837873085275428'+  comment: The hard-hexagon entropy constant, OEIS A085851 CITE{OEIShh}, from Baxter's+    exact solution CITE{Baxter80} evaluated at activity $z=1$; an algebraic number+    of degree $24$ CITE{Joyce}, the root of the polynomial CITE{formula-hard-hexagon}.+    Baxter gives $55$ decimals CITE{Baxter99}; the last digit listed in A085851 is+    one too small.+- params:+    model: hard-core+    lattice: triangular+    expression: entropy+  number: '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 CITE{Baxter99}.+- params:+    model: hard-core+    lattice: honeycomb+    expression: kappa+  number: '1.54644070878756141848902270530472278'+  comment: Baxter's corner-transfer-matrix value CITE{Baxter99}, printed there to+    $38$ decimals of which "the last two or three digits should be treated with caution",+    so $35$ are kept.+- params:+    model: hard-core+    lattice: honeycomb+    expression: entropy+  number: '0.4359559734410476815991032706617904'+  comment: $h=\ln\kappa$ from the $35$ decimals kept of Baxter's value CITE{Baxter99}.+- params:+    model: ice+    lattice: square+    expression: kappa+  number: '1.539600717839002038691063414671886548393604670053671669382939537290607126141155588516574388228665401'+  comment: $\kappa=\left(\frac43\right)^{3/2}=\frac{8\sqrt3}{9}$, Lieb's square ice+    constant CITE{Lieb}, OEIS A118273 CITE{OEISice}; the residual entropy of square+    ice, and the growth rate of the proper $3$-colourings of the square lattice.+- params:+    model: ice+    lattice: square+    expression: entropy+  number: '0.4315231086776713911588285089907411472552645663466415847599985280239394260811706965071663487686579294'+  comment: $h=\frac32\ln\frac43$ CITE{Lieb}.+- params:+    model: dimer+    lattice: square+    expression: kappa+  number: '1.338515151976096766938195902018513537064353697127911314641234786622391133007980978646487384617744539'+  comment: $\kappa=e^{G/\pi}$ with $G$ Catalan's constant CITE{Kasteleyn} CITE{TemperleyFisher},+    OEIS A097469 CITE{OEISdimer}; the number of domino tilings per cell. Per dimer+    the constant is $\kappa^2=e^{2G/\pi}=1.7916228\ldots$, OEIS A130834.+- params:+    model: dimer+    lattice: square+    expression: entropy+  number: '0.2915609040308187801383844564683949188640661539858372702610015691117476368804388617266268243031340589'+  comment: $h=G/\pi$ CITE{Kasteleyn}, OEIS A143233 CITE{OEISdimerh}; $G=L(2,\chi_{-4})$+    is in the HREF{Values_of_Dirichlet_L-functions_at_positive_integers}[table of+    Dirichlet $L$-values].+- params:+    model: dimer+    lattice: triangular+    expression: kappa+  number: '1.535098483272734137069289006038665026948796856053784183294104972609604690818919684539242687550327784'+  comment: $\kappa=e^{h}$ with $h$ the Kasteleyn integral CITE{formula-dimer-triangular}+    of Fendley, Moessner and Sondhi CITE{FMS}; per dimer the constant is $\kappa^2=2.3565273\ldots$,+    OEIS A247548 CITE{OEISdimertri}.+- params:+    model: dimer+    lattice: triangular+    expression: entropy+  number: '0.4285945374649588653584255564520155133111245974494630836695954222555248414097766638980597225348640087'+  comment: $h=\frac{1}{16\pi^2}\int_{-\pi}^{\pi}\!\int_{-\pi}^{\pi}\ln\left(6+2\cos+    u+2\cos v+2\cos(u+v)\right)du\,dv$ CITE{FMS}, who give $0.4286$; the integral+    is evaluated here after integrating $v$ in closed form.+- params:+    model: dimer+    lattice: honeycomb+    expression: kappa+  number: '1.175311211772651215599813186791236659885532007387707742635421848799592912335156217526580001540879280'+  comment: $\kappa=e^{h}$; the number of lozenge tilings per vertex of the honeycomb+    lattice CITE{Kasteleyn63}.+- params:+    model: dimer+    lattice: honeycomb+    expression: entropy+  number: '0.1615329736097252570468182553619031970361209203902935080654342351805075564036349210418938045446856960'+  comment: $h=\frac{3\sqrt3}{8\pi}L(2,\chi_{-3})=\frac12 m(1+x+y)$, half the Mahler+    measure of $1+x+y$ CITE{Smyth}, and one tenth of the spanning-tree constant of+    the triangular lattice; $L(2,\chi_{-3})$ is in the HREF{Values_of_Dirichlet_L-functions_at_positive_integers}[table+    of Dirichlet $L$-values].+- params:+    model: spanning-tree+    lattice: square+    expression: kappa+  number: '3.209912300728157678629749481779905158748592124251834494874586005846102464162424020406676712151410887'+  comment: '$\kappa=e^{4G/\pi}$, OEIS A229728 CITE{OEISste}: spanning trees per vertex+    of the square lattice.'+- params:+    model: spanning-tree+    lattice: square+    expression: entropy+  number: '1.166243616123275120553537825873579675456264615943349081044006276446990547521755446906507297212536236'+  comment: $z_{\mathrm{sq}}=\frac{4G}{\pi}=\frac4\pi\left(1-\frac1{3^2}+\frac1{5^2}-\cdots\right)$+    CITE{Wu77} CITE{ShrockWu}, OEIS A218387 CITE{OEISst}; the Mahler measure of $4+x+x^{-1}+y+y^{-1}$+    CITE{Guttmann}.+- params:+    model: spanning-tree+    lattice: triangular+    expression: kappa+  number: '5.029546072970906422186745870406623973669638516279899198827902912273509836816879007458803733878244240'+  comment: $\kappa=e^{z_{\mathrm{tri}}}$.+- params:+    model: spanning-tree+    lattice: triangular+    expression: entropy+  number: '1.615329736097252570468182553619031970361209203902935080654342351805075564036349210418938045446856960'+  comment: $z_{\mathrm{tri}}=\frac{3\sqrt3}{\pi}\left(1-\frac1{5^2}+\frac1{7^2}-\frac1{11^2}+\frac1{13^2}-\cdots\right)=\frac{15\sqrt3}{4\pi}L(2,\chi_{-3})$+    CITE{Wu77} CITE{ShrockWu}, OEIS A245725 CITE{OEISsttri}.+- params:+    model: spanning-tree+    lattice: honeycomb+    expression: kappa+  number: '2.242664948888020237028773931912807542795631222873299251361962513256207274076752068017801783089205771'+  comment: $\kappa=e^{z_{\mathrm{hc}}}$.+- params:+    model: spanning-tree+    lattice: honeycomb+    expression: entropy+  number: '0.8076648680486262852340912768095159851806046019514675403271711759025377820181746052094690227234284802'+  comment: $z_{\mathrm{hc}}=\frac12 z_{\mathrm{tri}}$ CITE{ShrockWu}, by the duality+    CITE{formula-duality}, OEIS A245737 CITE{OEISsthc}.+- params:+    model: spanning-tree+    lattice: kagome+    expression: kappa+  number: '3.113340924279582590607529456410547982636362178684166191368668056189257750714216803035170364118389361'+  comment: $\kappa=e^{z_{\mathrm{kag}}}$.+- params:+    model: spanning-tree+    lattice: kagome+    expression: entropy+  number: '1.135696401775102523760219970666578081028066632028646595503238898311987826408217630966139042419002579'+  comment: $z_{\mathrm{kag}}=\frac13\left(z_{\mathrm{tri}}+\ln 6\right)$ CITE{ShrockWu},+    OEIS A245739 CITE{OEISstkag}.+- params:+    model: spanning-tree+    lattice: D-3-6-3-6+    expression: kappa+  number: '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+    CITE{ShrockWu}.'+- params:+    model: spanning-tree+    lattice: D-3-6-3-6+    expression: entropy+  number: '1.135696401775102523760219970666578081028066632028646595503238898311987826408217630966139042419002579'+  comment: $z_{\mathrm{diced}}=z_{\mathrm{kag}}$ CITE{ShrockWu}, by the duality CITE{formula-duality}.+- params:+    model: spanning-tree+    lattice: 3-12-12+    expression: kappa+  number: '2.055590845879885707113290176216496540146858958934287667467234650725810280795254675322108282364948614'+  comment: $\kappa=e^{z}$ with $z=\frac16\left(z_{\mathrm{tri}}+\ln 15\right)$ CITE{ShrockWu}.+- params:+    model: spanning-tree+    lattice: 3-12-12+    expression: entropy+  number: '0.7205633228665771060773645206279575524223835193323670423836140961527914741604359903204479463922947767'+  comment: $z=\frac16\left(z_{\mathrm{tri}}+\ln 15\right)$ CITE{ShrockWu}, who give+    $0.7205633$.+- params:+    model: spanning-tree+    lattice: 4-8-8+    expression: kappa+  number: '2.196102669202442160697401841020442464994198552685921934118164015223723179250658218268867822440267179'+  comment: $\kappa=e^{z}$ with $z$ the integral CITE{formula-spanning-4-8-8} of Shrock+    and Wu CITE{ShrockWu}.+- params:+    model: spanning-tree+    lattice: 4-8-8+    expression: entropy+  number: '0.7866842753788321791216579894946953805511708165780327497186464518988179928818399372439686672615234781'+  comment: $z=\frac14\ln2+\frac1{4\pi}\int_0^{\pi}\ln\left(7-3\cos\theta+4\sin\frac\theta2\sqrt{5-\cos\theta}\right)d\theta$+    CITE{ShrockWu}, who give $0.786684(1)$.+- params:+    model: spanning-tree+    lattice: D-4-8-8+    expression: kappa+  number: '4.822866933678091099892364997288847432524076303535982947877790130594101785596674152160008146458081748'+  comment: $\kappa=e^{z}$ with $z$ twice the constant of $(4,8^2)$, its planar dual+    CITE{ShrockWu}.+- params:+    model: spanning-tree+    lattice: D-4-8-8+    expression: entropy+  number: '1.573368550757664358243315978989390761102341633156065499437292903797635985763679874487937334523046956'+  comment: $z_{\mathrm{UJ}}=2z_{(4,8^2)}$ CITE{ShrockWu}, by the duality CITE{formula-duality};+    they give $1.573368(2)$.+- params:+    model: spanning-tree+    lattice: sc+    expression: kappa+  number: '5.330202889205167421134597996649659520108'+  comment: $\kappa=e^{z_{\mathrm{sc}}}$, spanning trees per vertex of the simple cubic+    lattice.+- params:+    model: spanning-tree+    lattice: sc+    expression: entropy+  number: '1.673389302970196732283430621655598075258'+  comment: $z_{\mathrm{sc}}=\frac1{\pi^2}\int_0^{\pi}\!\int_0^{\pi}\operatorname{arccosh}(3-\cos\theta_1-\cos\theta_2)\,d\theta_1\,d\theta_2$+    CITE{formula-spanning-cubic}; the value $1.6741481(1)$ printed by Shrock and Wu+    CITE{ShrockWu} differs from it in the fourth decimal, and the closed-walk series+    $\ln 6-\sum_{m\geq1}W_{2m}/(2m\cdot 36^m)$ gives $1.6733893029701967322834\ldots$.