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：我们证明了角多面体和3连通Schnyder标号加入到可通过Kenyon、Miller、Sheffield和Wilson提出的双射与象限游走（加权）模型建立精确对应的平面结构之列。我们的方法首次给出多项式时间算法来计数这些结构，并确定其精确渐近增长常数：大小为$n$的角多面体数目$p_n$与3连通Schnyder标号数目$s_n$分别满足当$n$趋于无穷时$(p_n)^{1/n}\to 9/2$和$(s_n)^{1/n}\to 16/3$。尽管增长率为有理数（与此类对应中先前已知实例的情况一致），但指数增长的多项式修正项的指数似乎不遵循现今标准的Denisov-Wachtel方法，这是由于底层串联游走步集的双峰行为所致。然而，启发式论证表明这些指数对于$p_n$为$-1-\pi/\arccos(9/16)\approx -4.23$，对于$s_n$为$-1-\pi/\arccos(22/27)\approx -6.08$，这意味着相关级数不是D-有限的。