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Let $p_n$ denote the number of self-avoiding polygons of length $n$ on a regular three-dimensional lattice, and let $p_n(K)$ be the number which have knot type $K$. The probability that a random polygon of length $n$ has knot type $K$ is $p_n(K)/p_n$ and is known to decay exponentially with length. Little is known rigorously about the asymptotics of $p_n(K)$, but there is substantial numerical evidence that $p_n(K)$ grows as $p_n(K) simeq , C_K , mu_emptyset^n , n^{alpha-3+N_K}$, as $n to infty$, where $N_K$ is the number of prime components of the knot type $K$. It is believed that the entropic exponent, $alpha$, is universal, while the exponential growth rate, $mu_emptyset$, is independent of the knot type but varies with the lattice. The amplitude, $C_K$, depends on both the lattice and the knot type. The above asymptotic form implies that the relative probability of a random polygon of length $n$ having prime knot type $K$ over prime knot type $L$ is $frac{p_n(K)/p_n}{p_n(L)/p_n} = frac{p_n(K)}{p_n(L)} simeq [ frac{C_K}{C_L} ]$. In the thermodynamic limit this probability ratio becomes an amplitude ratio; it should be universal and depend only on the knot types $K$ and $L$. In this letter we examine the universality of these probability ratios for polygons in the simple cubic, face-centered cubic, and body-centered cubic lattices. Our results support the hypothesis that these are universal quantities. For example, we estimate that a long random polygon is approximately 28 times more likely to be a trefoil than be a figure-eight, independent of the underlying lattice, giving an estimate of the intrinsic entropy associated with knot types in closed curves.
We study the probability distribution P(M) of the order parameter (average magnetization) M, for the finite-size systems at the critical point. The systems under consideration are the 3-dimensional Ising model on a simple cubic lattice, and its 3-sta
A well known connection between first-passage probability of random walk and distribution of electrical potential described by Laplace equation is studied. We simulate random walk in the plane numerically as a discrete time process with fixed step le
We present a self-contained discussion of the universality classes of the generalized epidemic process (GEP) on Poisson random networks, which is a simple model of social contagions with cooperative effects. These effects lead to rich phase transitio
Recently a nonuniversal character of the leading spatial behavior of the thermodynamic Casimir force has been reported [X. S. Chen and V. Dohm, Phys. Rev. E {bf 66}, 016102 (2002)]. We reconsider the arguments leading to this observation and show tha
Master equations are common descriptions of mesoscopic systems. Analytical solutions to these equations can rarely be obtained. We here derive an analytical approximation of the time-dependent probability distribution of the master equation using ort