可平面图的圈率与沙堆密度
The looping rate and sandpile density of planar graphs
摘要：
给出了有限可平面图上环擦除随机游动的环率的一个简单公式。循环速率与图上循环沙堆中的预期砂量密切相关。循环率公式非常适合于在图趋于无限格的情况下取极限，并且我们使用它来给出（先前计算的）正方形、三角形和蜂窝晶格的循环速率和沙堆密度，并计算（第一次）许多其他晶格的循环率和沙堆密度，例如Kagome晶格，骰子晶格，截顶六边形晶格（其值都是有理数）和正方形-八边形晶格（其值是超越的）。
英文摘要：
We give a simple formula for the looping rate of loop-erased random walk on a finite planar graph. The looping rate is closely related to the expected amount of sand in a recurrent sandpile on the graph. The looping rate formula is well-suited to taking limits where the graph tends to an infinite lattice, and we use it to give an elementary derivation of the (previously computed) looping rate and sandpile densities of the square, triangular, and honeycomb lattices, and compute (for the first time) the looping rate and sandpile densities of many other lattices, such as the kagome lattice, the dice lattice, and the truncated hexagonal lattice (for which the values are all rational), and the square-octagon lattice (for which it is transcendental).
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
