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The pretty good transmission scenario implies that the probability of sendingone excitation from one extreme of a spin chain to the other can reach valuesarbitrarily close to the unity just by waiting a time long enough. Theconditions that ensure the appearance of this scenario are known for chainswith different interactions and lengths. Sufficient conditions for the presenceof pretty good transmission depend on the spectrum of the Hamiltonian of thespin chain. Some works suggest that the time $t_{\varepsilon}$ at which thepretty good transmission takes place scales as $1/(|\varepsilon|)^{f(N)}$,where $\varepsilon$ is the difference between the probability that a singleexcitation propagates from one extreme of the chain to the other and the unity,while $f(N)$ is an unknown function of the chain length. In this paper, we showthat the exponent is not a simple function of the chain length but a power lawof the number of linearly independent irrational eigenvalues of theone-excitation block of the Hamiltonian that enter into the expression of theprobability of transmission of one excitation. We explicitly provide examplesof a chain showing that the exponent changes when the couplings between thespins change while the length remains fixed. For centrosymmetric spin chainsthe exponent is at most $N/2$.

The scaling law of the arrival time of spin systems that present pretty good transmission