History of Connective constants of lattices

 Display properties:   number-header: $\mu$-Numbers: []+Numbers:+- params:+    lattice: honeycomb+    d: '2'+  number: '1.847759065022573512256366378793576573644833251727284972230195462561070015002204717429679869700689192'+  comment: '$\mu=\sqrt{2+\sqrt2}=2\cos\frac{\pi}{8}$, exact: conjectured by Nienhuis+    CITE{Nienhuis} and proved by Duminil-Copin and Smirnov CITE{DuminilCopinSmirnov};+    the largest root of $x^4-4x^2+2$, OEIS A179260 CITE{OEIShoneycomb}; $\cos\frac{\pi}{8}$+    is in the HREF{Cos_pi_times_x_at_rational_numbers#1/8}[table of $\cos(\pi x)$];+    the honeycomb lattice is also called the hexagonal lattice.'+- params:+    lattice: square+    d: '2'+  number: 2.63815853032790 +/- 3e-14+  comment: $2.63815853032790(3)$ CITE{JSG16}, from the topological transfer matrix;+    the conjecture of Guttmann that $\mu$ is the positive root of $13x^4-7x^2-581$,+    which is $2.63815853034\ldots$ CITE{OEISconjecture}, fails in the twelfth digit;+    the first thirteen digits are OEIS A387897 CITE{OEISsquare}.+- params:+    lattice: triangular+    d: '2'+  number: 4.150797226 +/- 26e-9+  comment: $4.150797226(26)$ CITE{Jensen04}, 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.+- params:+    lattice: kagome+    d: '2'+  number: 2.560576765 +/- 10e-9+  comment: $2.560576765(10)$ CITE{Jensen04bounds}, from unpublished enumerations of+    self-avoiding polygons; the earlier $2.56062$ of Jensen and Guttmann CITE{JensenGuttmann98}+    is the value listed in CITE{Wiki}.+- params:+    lattice: 3-12-12+    d: '2'+  number: '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 CITE{OEIStt}: 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+    CITE{JensenGuttmann98} CITE{GuttmannParviainenRechnitzer}; the lattice is also+    called the three-twelve or truncated hexagonal lattice.'+- params:+    lattice: 4-8-8+    d: '2'+  number: 1.80883001 +/- 6e-8+  comment: $1.80883001(6)$ CITE{JensenGuttmann98}, from series analysis, as quoted+    in CITE{Jensen04bounds} and CITE{Alm}; the lattice is also called the four-eight,+    bathroom-tile or truncated square lattice.+- params:+    lattice: manhattan+    d: '2'+  number: 1.733535 +/- 2e-6+  comment: $1.733535(2)$ CITE{BennettWood98}, from series for self-avoiding polygons+    of up to 84 steps; the table in CITE{Wiki} lists the uncertainty as $3$ in the+    last digit.+- params:+    lattice: L+    d: '2'+  number: 1.5657 +/- 15e-4+  comment: $1.5657(15)$ as listed in CITE{Wiki}, which names no source for this value;+    no paper stating it could be read for this table, and the walk counts of OEIS+    A322419 CITE{OEISL} are consistent with it.+- params:+    lattice: sc+    d: '3'+  number: 4.684039931 +/- 27e-9+  comment: $4.684039931(27)$ CITE{Clisby13}, from the pivot algorithm; the enumeration+    of Schram, Barkema and Bisseling CITE{SchramBarkemaBisseling} gives $4.684043(12)$.+- params:+    lattice: bcc+    d: '3'+  number: 6.530511501 +/- 84e-9+  comment: $6.530511501(84)$ CITE{Clisby22}, from the pivot algorithm; the enumeration+    of Schram, Barkema, Bisseling and Clisby CITE{SchramEtAl17} gives $6.530520(20)$.+- params:+    lattice: fcc+    d: '3'+  number: 10.03705785 +/- 14e-8+  comment: $10.03705785(14)$ CITE{Clisby22}, from the pivot algorithm; the enumeration+    of Schram, Barkema, Bisseling and Clisby CITE{SchramEtAl17} gives $10.037075(20)$.+- params:+    lattice: hypercubic+    d: '4'+  number: 6.774043 +/- 5e-6+  comment: $6.774043(5)$ CITE{OwczarekPrellberg}, from the pivot algorithm.+- params:+    lattice: hypercubic+    d: '5'+  number: 8.838544 +/- 3e-6+  comment: $8.838544(3)$ CITE{OwczarekPrellberg}, from the pivot algorithm.+- params:+    lattice: hypercubic+    d: '6'+  number: 10.878094 +/- 4e-6+  comment: $10.878094(4)$ CITE{OwczarekPrellberg}, from the pivot algorithm.+- params:+    lattice: hypercubic+    d: '7'+  number: 12.902817 +/- 3e-6+  comment: $12.902817(3)$ CITE{OwczarekPrellberg}, from the pivot algorithm.+- params:+    lattice: hypercubic+    d: '8'+  number: 14.919257 +/- 2e-6+  comment: $14.919257(2)$ CITE{OwczarekPrellberg}, from the pivot algorithm.