Clisby 2007 self avoiding walk enumeration via lace

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enumeration_results: Clisby, Liang and Slade's exact enumerations on Z^d: polygons p_n for n <= 32 in d = 3, n <= 26 in d = 4 and n <= 24 in every d >= 5, and walks c_n with the moments rho_n for n <= 30 in d = 3 and n <= 24 in every d >= 4, including c_30 = 270569905525454674614 on Z^3.

equation_1: Clisby, Liang and Slade's expansion of the connective constant mu of the self-avoiding walk on Z^d in powers of 1/(2d), from 2d - 1 - 1/(2d) through the term of order (2d)^{-11} with an error O((2d)^{-12}), seven of its coefficients new and the error estimate stated to be rigorous.

equations_3_4: Clisby, Liang and Slade's expansions in powers of 1/(2d) of the amplitudes A and D in c_n = A mu^n [1 + O(n^{-epsilon})] and mean-square displacement D n [1 + O(n^{-epsilon})] for d >= 5, each through order (2d)^{-12} with an error O((2d)^{-13}).

section_7_2: Clisby, Liang and Slade's upper bounds on the connective constant mu of Z^d for d = 3, ..., 12, from 4.7552 for d = 3 to 22.9549 for d = 12, obtained by inserting their exact walk counts into the Ahlberg-Janson bound; for d = 3 to 6 these are weaker than bounds already known.

theorem_2_1: Clisby, Liang and Slade's formula for the weight of a 2-step walk, the number of 2n-step self-avoiding walks whose every second vertex traces it: zero if a component of its allocation graph has two or more loops or cycles, and otherwise 2 per one-cycle component times the vertex count of each tree component.

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Clisby, Nathan and Liang, Richard and Slade, Gordon, Self-avoiding walk enumeration via the lace expansion. J. Phys. A: Math. Theor. 40 (2007), 10973-11017. DOI 10.1088/1751-8113/40/36/003.

The paper introduces a new method for enumerating self-avoiding walks (SAWs) based on the lace expansion, which rewrites the number of n-step SAWs in terms of counts of lace graphs (self-avoiding polygons and related self-intersecting trajectories) that are far less numerous, plus an algorithmic device called the two-step method giving an exponential improvement over brute force. With these they enumerate 32-step self-avoiding polygons in d = 3, 26-step polygons in d = 4, 30-step SAWs in d = 3, and 24-step walks and polygons in all d >= 4; explicit values include c_30 = 270569905525454674614 and p_32 = 53424552150523386 on the cubic lattice, with tables for d = 3, 4, 5, 6 in Appendix A (pp. 44-48) and more in the authors' companion tables. They note the structural fact that enumerating c_n for n <= 2k and d <= k determines c_n for n <= 2k in all dimensions. Series analysis then yields numerical estimates of the connective constant in dimensions 3 <= d <= 8, of critical amplitudes, and of critical exponents for d = 3. Separately, the lace-graph counts combined with lace-expansion bounds give substantially extended 1/d expansions for the connective constant and two critical amplitudes, whose error estimates the paper states are rigorous. Section 7.2 inserts the new counts into an upper bound it credits to Ahlberg and Janson, obtaining upper bounds on the connective constant for 3 <= d <= 12 that the paper says are weaker, for d = 3 to 6, than bounds already known. For problem 528, which concerns the number of self-avoiding walks and the connective constant, this supplies exact counts c_n for n <= 30 in d = 3 and n <= 24 in every dimension d >= 4, numerical estimates of the connective constant for 3 <= d <= 8 from series analysis (estimates, not proved bounds), and the 1/d expansion of the connective constant.

Source: https://personal.math.ubc.ca/~slade/research.html. The copy read for this card is the author's manuscript dated July 24, 2007, which prints no notice, from the author's research page (https://personal.math.ubc.ca/~slade/research.html), which states no terms; the term is unstated.

Read status. Claims checked: equations (1), (3) and (4), the enumeration ranges and displayed values of Section 1.2, the reduction of Section 3.3, Theorem 2.1 and the bounds of Section 7.2 were read clause by clause on the printed pages. The proof of the error estimates (Section 4.3, pp. 20-23) was read but not checked step by step, and the computer enumerations were not reproduced. Pages cited are the manuscript's.

Bears on. #528: the problem's C_k is the paper's mu with d = k. Equation (1) gives the asymptotics of C_k as k tends to infinity through order (2k)^{-11}, with remainder O((2k)^{-12}); Section 7.2 gives upper bounds on C_k for 3 <= k <= 12, derived from an inequality the paper states is proved in an unpublished manuscript and, for k = 3 to 6, weaker than bounds already known; the enumerations give exact values of the problem's f(n, k) for n <= 30 when k = 3 and n <= 24 for every k >= 4. None of these determines C_k for any fixed k, and the paper's series-analysis values of mu for 3 <= d <= 8 are numerical estimates, not bounds.

Results. [[discrete_geometry/clisby_2007_self_avoiding_walk_enumeration_via_lace/enumeration_results|Section 1.2]] (p. 3, with the reduction of Section 3.3, p. 15, and Appendix A, pp. 44-48); [[discrete_geometry/clisby_2007_self_avoiding_walk_enumeration_via_lace/equation_1|Equation (1)]] (p. 3); [[discrete_geometry/clisby_2007_self_avoiding_walk_enumeration_via_lace/equations_3_4|Equations (3) and (4)]] (p. 4); [[discrete_geometry/clisby_2007_self_avoiding_walk_enumeration_via_lace/theorem_2_1|Theorem 2.1]] (p. 7); [[discrete_geometry/clisby_2007_self_avoiding_walk_enumeration_via_lace/section_7_2|Section 7.2]] (p. 44). The series analysis of Sections 5, 6 and 7.1 yields numerical estimates and has no result page.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.