arXiv 1101.40472011-01cond-mat.stat-mech

纽结概率比的普适性

On the universality of knot probability ratios

E. J. Janse van Rensburg, A. Rechnitzer

备注：: 8 pages, 6 figures, 1 table
DOI：: 10.1088/1751-8113/44/16/162002

摘要：

令$p_n$表示规则三维格上长度为$n$的自回避多边形的数目，并且令$p_n(K)$表示具有结类型$K$的数目。长度为$n$的随机多边形具有结类型$K$的概率为$p_n(K)/p_n$，并且已知其随长度呈指数衰减。关于$p_n(K)$的渐近性我们知之甚少，但是有大量的数值证据表明$p_n(K)$以$p_n(K) \simeq \, C_K \, μ_\emptyset^n \, n^{α-3+N_K}$的形式增长，以$n \to \infty$的形式增长，其中$N_K$是结型$K$的素分量的个数。我们认为熵指数$α$是普适的，而指数增长率$μ_\emptyset$与结的类型无关，但随晶格而变化。振幅$C_K$取决于晶格和结类型。上述渐近形式意味着具有素数结类型$K$的长度为$n$的随机多边形相对于素数结类型$L$的相对概率为$\frac{p_n(K)/p_n}{p_n(L)/p_n} = \frac{p_n(K)}{p_n(L)} \simeq [ \frac{C_K}{C_L} ]$。在热力学极限下，这个概率比变成振幅比;它应该是普适的，并且只取决于结类型$K$和$L$。在这封信中，我们研究这些概率比的多边形在简单的立方，面心立方和体心立方晶格的普遍性。我们的研究结果支持这一假设，即这些是普遍的量。例如，我们估计一个长的随机多边形是三叶形的可能性大约是8的28倍，与底层晶格无关，给出了与封闭曲线中的结类型相关的固有熵的估计。

英文摘要：

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 $p_n(K) \simeq \, C_K \, μ_\emptyset^n \, n^{α-3+N_K}$, 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, $α$, is universal, while the exponential growth rate, $μ_\emptyset$, 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.

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