Table 1 Summary of the time dependence of F_n(τ) for the three classes of RWs—recurrent, marginal, and transient

	t_n	T_n	1 ≪ τ ≪ t_n	t_n ≪ τ ≪ T_n	T_n ≪ τ
μ < 1 [recurrent]	n^1/μ	n^1/μ	τ^−(1+μ) ≡ τ^−(2−θ)		\(\exp \left[-{{{{{{{\rm{const}}}}}}}}\,\tau /{n}^{1/\mu }\right]\)
μ = 1 [marginal]	\(\sqrt{n}\)	n^3/2	τ^-⁽¹⁺^μ⁾≡τ^-⁽²^-^θ⁾	\(\exp \big[-{{{{{{{\rm{const}}}}}}}}{\left(\tau /{t}_{n}\right)}^{\mu /(1+\mu )}\big]\)	\(\exp \left[-{{{{{{{\rm{const}}}}}}}}\,\tau /{n}^{1/\mu }\right]\)
μ > 1 [transient]	1	n^(μ+1)/μ		\(\exp \big[-{{{{{{{\rm{const}}}}}}}}{\left(\tau /{t}_{n}\right)}^{\mu /(1+\mu )}\big]\)	\(\exp \left[-{{{{{{{\rm{const}}}}}}}}\,\tau /{n}^{1/\mu }\right]\)

The constants are independent of n and τ. The crossover times t_n and T_n are given up to logarithmic prefactors. The time regimes identified in the last three columns are the same as the ones presented in Fig. 2. The persistence exponent θ is here given by θ = 1 − μ, see text.

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