Scaling Exponents for Ordered Maxima
E. Ben-Naim and P.L. Krapivsky
We study extreme value statistics of multiple sequences of random
variables. For each sequence with $N$ variables, independently
drawn from the same distribution, the running maximum is defined as
the largest variable to date. We compare the running maxima of $m$
independent sequences, and investigate the probability $S_N$ that
the maxima are perfectly ordered, that is, the running maximum of
the first sequence is always larger than that of the second
sequence, which is always larger than the running maximum of the
third sequence, and so on. The probability $S_N$ is universal: it
does not depend on the distribution from which the random variables
are drawn. For two sequences, $S_N\sim N^{-1/2}$, and in general,
the decay is algebraic, $S_N\sim N^{-\sigma_m}$, for large $N$. We
analytically obtain the exponent $\sigma_3\cong 1.302931$ as root of
a transcendental equation. Furthermore, the exponents $\sigma_m$
grow with $m$, and we show that $\sigma_m \sim m$ for large $m$.
source,
pdf