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The polarization process of conventional polar codes in binary erasurechannel (BEC) is recast to the Domany-Kinzel cellular automaton model ofdirected percolation in a tilted square lattice. Consequently, the former'sscaling exponent, $\mu$, can be analogously expressed as the inverse of thepercolation critical exponent, $\beta$. Relying on the vast percolation theoryliterature and the best known numerical estimate for $\beta$, the scalingexponent can be easily estimated as$\mu_{\text{num}}^{\text{perc}}\simeq1/0.276486(8)\simeq3.617$, which is onlyabout $0.25\%$ away from the known exponent computation from coding theoryliterature based on numerical approximation, $\mu_{\text{num}}\simeq3.627$.Remarkably, this numerical result for the critical exponent, $\beta$, can beanalytically approximated (within only $0.028\%$) leading to the closed-formexpression for the scaling exponent$\mu\simeq2+\varphi=2+1.618\ldots\simeq3.618$, where$\varphi\triangleq(1+\sqrt{5})/2$ is the ubiquitous golden ratio. As theultimate achievable scaling exponent is quadratic, this implies that thepenalty for polar codes in BEC, in terms of the scaling exponent, can be verywell estimated by the golden ratio, $\varphi$, itself.

The Penalty in Scaling Exponent for Polar Codes is Analytically Approximated by the Golden Ratio