We show that corner polyhedra and 3-connected Schnyder labelings join the growing list of planar structures that can be set in exact correspondence with (weighted) models of quadrant walks via a bijection due to Kenyon, Miller, Sheffield and Wilson. Our approach leads to a first polynomial time algorithm to count these structures, and to the determination of their exact asymptotic growth constants: the number $p_n$ of corner polyhedra and $s_n$ of 3-connected Schnyder labelings of size $n$ respectively satisfy $(p_n)^{1/n}\to 9/2$ and $(s_n)^{1/n}\to 16/3$ as $n$ goes to infinity. While the growth rates are rational, like in the case of previously known instances of such correspondences, the exponent of the asymptotic polynomial correction to the exponential growth does not appear to follow from the now standard Denisov-Wachtel approach, due to a bimodal behavior of the step set of the underlying tandem walk. However a heuristic argument suggests that these exponents are $-1-\pi/\arccos(9/16)\approx -4.23$ for $p_n$ and $-1-\pi/\arccos(22/27)\approx -6.08$ for $s_n$, which would imply that the associated series are not D-finite.

翻译：我们证明角多面体与3-连通Schnyder标号可归入一类不断增长的平面结构族，这类结构通过Kenyon、Miller、Sheffield和Wilson提出的双射与（加权）象限游走模型建立精确对应。我们的方法首次给出计数这些结构的多项式时间算法，并确定了其精确渐近增长常数：规模为n的角多面体数量p_n与3-连通Schnyder标号数量s_n分别满足n→∞时(p_n)^{1/n}→9/2和(s_n)^{1/n}→16/3。尽管增长率与先前已知的此类对应案例一样为有理数，但由于底层串联游走的步集呈现双峰特征，渐近多项式修正项的指数似乎无法通过当前标准的Denisov-Wachtel方法获得。然而启发式论证表明，p_n的该指数约为-1-π/arccos(9/16)≈-4.23，s_n的约为-1-π/arccos(22/27)≈-6.08，这意味着相关级数非D-有限。