Rank distributions of words in correlated symbolic systems and the Zipf law

The binary many-step Markov chain with the steplike memory function is considered as a model for the analysis of rank distributions of words in correlated stochastic symbolic systems. We prove that this distribution obeys the power law with the exponent of the order of unity in the case of rather strong persistent correlations. The Zipf law is shown to be valid for the rank distribution of words with lengths about and shorter than the correlation length in the Markov sequence. A self-similarity in the rank distribution with respect to the decimation procedure is observed.

Figure 1

The rank distribution $W\left(R\right)$ of the words with length $L=14$ in the Markov chain determined by the conditional probability Eq. (5) at $N=15$, $\mu =15∕46$. The solid line is for the numerical simulations, the dotted line describes the exact distribution obtained from Eqs. (7, 9, 10), the dash-dotted line is the plot of envelope asymptotics (11), the dash-dot-dotted line describes the Zipf’s-like asymptotics Eq. (13) with $\zeta =7∕11$. The thin solid line obtained numerically at $\mu =-15∕46$, $N=15$, $L=14$ is for the antipersistent correlations.

Figure 2

Rank distributions of the $L$ words with $L=14$ in the $N$-step Markov sequences reduced by randomly removing some symbols. The solid line is for the initial sequence with $N=32$, $n=15$. The symbols corresponding to the decimation parameter $\lambda =2$ $(N^{*}=16)$ lie almost on the solid one. Other lines are for decimated sequences with $N^{*}<L$. Specifically, the dash-dotted, dash-dot-dotted, and dashed lines correspond to $N^{*}=8$, $N^{*}=4$, and to $N^{*}=2$, respectively.

Figure 3

The Zipf plots for $L$ words with $L>N$. The solid, dash-dotted, and dashed lines are for $L=12,14,18$, respectively. The parameters of the Markov chain are $N=12$, $\mu =0.1$. The phenomenon of step smearing is noted with a growth in the word length $L$. In the inset: the Zipf plots for $L$ words with $N=4$, $\mu =0.1$ at $L>L_{\mathit{cr}}$ (the lengths of words are shown near the curves).