椭圆上的密切随机游动
Osculating Random Walks on Cylinders
摘要：
我们考虑正方形格上的随机路径，它们在每个顶点处都左转或右转。除了在之前已经访问过的顶点处之外，可能的转弯以相等的概率被采取。在这种情况下，顶点通过未使用的边离开。当到达初始边缘时，认为路径完成。我们还考虑家庭的这种道路一起覆盖每一个边缘的格一次，并访问每个顶点两次。因为这些路径可能彼此接触但不相交，我们称它们为密切步行。这些族的系综也被称为稠密O $(n=1)$模型。我们特别考虑这样的路径在一个圆柱形的几何形状，与圆柱轴平行的晶格方向之一。我们制定了一个猜想的概率，一个面对的晶格是由$m$不同的密切路径包围。甚至系统的大小，我们给出了一个猜想的概率，一条道路的风周围的圆柱。对于奇数系统的大小，我们推测的概率，一个点被访问的路径跨越无限长的圆柱。最后，我们猜想一个二项式行列式的渐近表达式。
英文摘要：
We consider random paths on a square lattice which take a left or a right turn at every vertex. The possible turns are taken with equal probability, except at a vertex which has been visited before. In such case the vertex is left via the unused edge. When the initial edge is reached the path is considered completed. We also consider families of such paths which together cover every edge of the lattice once and visit every vertex twice. Because these paths may touch but not intersect each other and themselves, we call them osculating walks. The ensemble of such families is also known as the dense O$(n=1)$ model. We consider in particular such paths in a cylindrical geometry, with the cylindrical axis parallel with one of the lattice directions. We formulate a conjecture for the probability that a face of the lattice is surrounded by $m$ distinct osculating paths. For even system sizes we give a conjecture for the probability that a path winds round the cylinder. For odd system sizes we conjecture the probability that a point is visited by a path spanning the infinite length of the cylinder. Finally we conjecture an expression for the asymptotics of a binomial determinant.
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