History of Entropy constants of lattice models

back to table · edit · history · where entries came from · files

when	who	what
2026-09-10 09:21	bmatschke	the differential operator is upright; d is not a variable	current reviewed
2026-09-09 08:58	bmatschke	link the entry the sentence means, not the whole table
2026-09-07 01:45	bmatschke	tagged "statistical mechanics": six tables on critical phenomena carried only "physics" and "combinatorics", which does not distinguish them from anything
2026-09-06 13:33	bmatschke	the arXiv number of 5 references was in the sentence, where it is text; moved to the `arxiv` field, which the page renders as a link to the abstract
2026-09-06 05:59	zeta3 with Claude Code, table-repair@c87022a	after the critique: the document in the repository order again, parameters model, lattice, expression and the rows in the generator order; ten closed forms, not nine; Liang proves 25 decimals, not 27; the spanning-tree model named; the kagome, diced and (3,12^2) factors; A130834 linked; residual ent
2026-09-06 05:36	zeta3 with Claude Code, table-build@bc74763	after audit_table: the Definition is one sentence, the four models being named in the parameter and the models comment, and the golden ratio is linked where the status comment names it
2026-09-06 05:33	zeta3 with Claude Code, table-build@bc	entropy constants kappa and h = ln kappa per site of the hard-core, ice, dimer and spanning-tree models on lattices, 34 entries: closed forms and two integrals in ball arithmetic at 100 digits, the hard-hexagon constant from Baxter's exact solution, the simple-cubic spanning-tree constant by quadrat
2026-09-06 05:32	zeta3	checking that this table can be written to
2026-09-06 05:32	zeta3 with Claude Code, table-build@bc74763	draft: entropy constants of lattice models, prose first, entries to follow from generate.py

What changed between 2026-09-09 08:58 and 2026-09-10 09:21

from line 91 (8 lines) @@ -91,8 +91,8 @@

     j}{m+1}+4\cos^2\frac{\pi k}{n+1}\right)$.   formula-dimer-honeycomb: $h(\text{dimer},6^3)=\frac{1}{16\pi^2}\int_{-\pi}^{\pi}\!\int_{-\pi}^{\pi}\ln\left(3+2\cos-    u+2\cos v+2\cos(u+v)\right)du\,dv=\frac{3\sqrt3}{8\pi}L(2,\chi_{-3})$ CITE{Kasteleyn63},-    half the Mahler measure of $1+x+y$ CITE{Smyth}, and $h(\text{dimer},6^3)=\frac1{10}z_{\mathrm{tri}}$.+    u+2\cos v+2\cos(u+v)\right)\,\mathrm{d}u\,\mathrm{d}v=\frac{3\sqrt3}{8\pi}L(2,\chi_{-3})$+    CITE{Kasteleyn63}, half the Mahler measure of $1+x+y$ CITE{Smyth}, and $h(\text{dimer},6^3)=\frac1{10}z_{\mathrm{tri}}$.   formula-dimer-triangular: $h(\text{dimer},3^6)=\frac{1}{16\pi^2}\int_{-\pi}^{\pi}\!\int_{-\pi}^{\pi}\ln\left(6+2\cos-    u+2\cos v+2\cos(u+v)\right)du\,dv=\frac{1}{8\pi}\int_{-\pi}^{\pi}\ln\frac{A+\sqrt{A^2-B^2}}{2}\,du$+    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 $B=4\cos\frac u2$ CITE{FMS}.   formula-spanning-planar: $z_{\mathrm{sq}}=\frac{4G}{\pi}$, $z_{\mathrm{tri}}=\frac{3\sqrt3}{\pi}\left(1-\frac1{5^2}+\frac1{7^2}-\frac1{11^2}+\frac1{13^2}-\cdots\right)=\frac{15\sqrt3}{4\pi}L(2,\chi_{-3})$,

from line 103 (5 lines) @@ -103,5 +103,5 @@

   formula-spanning-4-8-8: $z_{(4,8^2)}=\frac14\ln2+\frac1{4\pi}\int_0^{\pi}\ln\left(7-3\cos\theta+4\sin\frac\theta2\sqrt{5-\cos\theta}\right)d\theta$     and $z_{\mathrm{UJ}}=2z_{(4,8^2)}$ for the union-jack lattice CITE{ShrockWu}.-  formula-spanning-cubic: $z_{\mathrm{sc}}=\ln6+\frac{1}{(2\pi)^3}\int_{[-\pi,\pi]^3}\ln\left(1-\frac{\cos\theta_1+\cos\theta_2+\cos\theta_3}{3}\right)d^3\theta=\frac1{\pi^2}\int_0^{\pi}\!\int_0^{\pi}\operatorname{arccosh}\left(3-\cos\theta_1-\cos\theta_2\right)d\theta_1\,d\theta_2=\ln6-\sum_{m\geq1}\frac{W_{2m}}{2m\cdot36^m}$+  formula-spanning-cubic: $z_{\mathrm{sc}}=\ln6+\frac{1}{(2\pi)^3}\int_{[-\pi,\pi]^3}\ln\left(1-\frac{\cos\theta_1+\cos\theta_2+\cos\theta_3}{3}\right)d^3\theta=\frac1{\pi^2}\int_0^{\pi}\!\int_0^{\pi}\operatorname{arccosh}\left(3-\cos\theta_1-\cos\theta_2\right)d\theta_1\,\mathrm{d}\theta_2=\ln6-\sum_{m\geq1}\frac{W_{2m}}{2m\cdot36^m}$     CITE{ShrockWu}, 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 $\mathbb{Z}^3$ CITE{OEISwalks}.

from line 379 (6 lines, 1 more than before) @@ -379,5 +379,6 @@

       entropy:         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 $0.4286$.+          u+2\cos v+2\cos(u+v)\right)\,\mathrm{d}u\,\mathrm{d}v$ CITE{FMS}, who give+          $0.4286$.         number: '0.4285945374649588653584255564520155133111245974494630836695954222555248414097766638980597225348640087'     honeycomb:

from line 471 (5 lines) @@ -470,5 +471,5 @@

         number: '5.330202889205167421134597996649659520108'       entropy:-        comment: $z_{\mathrm{sc}}=\frac1{\pi^2}\int_0^{\pi}\!\int_0^{\pi}\operatorname{arccosh}(3-\cos\theta_1-\cos\theta_2)\,d\theta_1\,d\theta_2$+        comment: $z_{\mathrm{sc}}=\frac1{\pi^2}\int_0^{\pi}\!\int_0^{\pi}\operatorname{arccosh}(3-\cos\theta_1-\cos\theta_2)\,\mathrm{d}\theta_1\,\mathrm{d}\theta_2$           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