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Asymptotics of the number of lattice triangulations of rectangles of width 4 and 5

Stepan Orevkov

TL;DR

This work analyzes the asymptotics of primitive lattice triangulations f(m,n) of m×n rectangles for m=4,5 by converting the counting problem into coupled systems of Fredholm integral equations on generating functions. The authors derive width-4 and width-5 formulations via trapezoid reductions, recurrences over polygonal shapes, and strategic variable changes, then discretize the integral equations to compute the limits c_4 and c_5 with high precision (and obtain new exact values for larger m). Crucially, they confront non-compact operators and employ convergence-acceleration techniques and hybrid MATLAB/Mathematica workflows to handle large-scale high-precision linear solves, enabling 65-digit accuracy for c_4 and 15 digits for c_5, as well as empirical estimates for c_6 and c_7. The results tighten lower bounds on the global capacity c, support a convexity conjecture for f(m,n), and provide a rigorous framework for extending the approach to larger widths, while also proving a non-primitive growth lower bound of 5 for f^{np}(n,n).

Abstract

Let $f(m,n)$ be the number of primitive lattice triangulations of an $m \times n$ rectangle. We express the limits $\lim_n f(m,n)^{1/n}$ for $m = 4$ and $m=5$ in terms of certain systems of Fredholm integral equations on generating functions (the case $m\le3$ was treated in a previous paper). Solving these equations numerically, we compute approximate values of these limits with a rather high precision.

Asymptotics of the number of lattice triangulations of rectangles of width 4 and 5

TL;DR

This work analyzes the asymptotics of primitive lattice triangulations f(m,n) of m×n rectangles for m=4,5 by converting the counting problem into coupled systems of Fredholm integral equations on generating functions. The authors derive width-4 and width-5 formulations via trapezoid reductions, recurrences over polygonal shapes, and strategic variable changes, then discretize the integral equations to compute the limits c_4 and c_5 with high precision (and obtain new exact values for larger m). Crucially, they confront non-compact operators and employ convergence-acceleration techniques and hybrid MATLAB/Mathematica workflows to handle large-scale high-precision linear solves, enabling 65-digit accuracy for c_4 and 15 digits for c_5, as well as empirical estimates for c_6 and c_7. The results tighten lower bounds on the global capacity c, support a convexity conjecture for f(m,n), and provide a rigorous framework for extending the approach to larger widths, while also proving a non-primitive growth lower bound of 5 for f^{np}(n,n).

Abstract

Let be the number of primitive lattice triangulations of an rectangle. We express the limits for and in terms of certain systems of Fredholm integral equations on generating functions (the case was treated in a previous paper). Solving these equations numerically, we compute approximate values of these limits with a rather high precision.

Paper Structure

This paper contains 22 sections, 5 theorems, 109 equations, 9 figures.

Table of Contents

  1. Introduction. Main results
  2. Strips of width 4
  3. Reduction to trapezoids (which decreases the dimension of the problem)
  4. Recurrence relations
  5. A change of variables
  6. Elimination of $f$, $h_1$, and $h_2$
  7. Estimates for the radii of convergence
  8. The system of integral equations and its discretization
  9. Strips of width 5
  10. Reduction to trapezoids
  11. Recurrence relations
  12. A change of variables
  13. Elimination of $f$, $h_1$, $h_2$, $j_1$, $j_2$, $j_4$
  14. Computation of $c_5$
  15. Computational issues
  16. ...and 7 more sections

Key Result

Lemma 2.1

All coefficients of $f$, $g_k$, $h_k$, $j_1$, $\Phi_k$, $1/q$, $1/p$, $1/r$, $1/(p\Psi_1)$, $1/(r\Psi_2)$, $1/\Psi_{3}$ are positive, and $R_x(1/q)=1/2$, $R_x(1/p)=(-1+\sqrt5)/2\approx 0.618$, $R_x(1/r)=1$.

Figures (9)

  • Figure 1:
  • Figure 2: The graphs of $\Psi_1(x;1,1)$, $\Psi_2(x;1,1)$, $\Psi_{3}(x;1)$ on $\left[\frac{1}{4},\frac{1}{2}\right]$.
  • Figure 3: The graph of $J(x)$.
  • Figure 4: Trapezoids that contribute to $\ell^*_{\lambda\mu}(3)$ and $\ell_{\lambda\mu}(3)$.
  • Figure 5: The shapes of $F_1, G_{1\nu}, H_{1\nu}, J_{1\nu}, L_{1\nu}$.
  • ...and 4 more figures

Theorems & Definitions (17)

  • Remark 1
  • Lemma 2.1
  • proof
  • Remark 2
  • Lemma 2.2
  • proof
  • Remark 3
  • Lemma 2.3
  • proof
  • Remark 4
  • ...and 7 more