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We study the probability density function $P(h_m,L)$ of the maximum relativeheight $h_m$ in a wide class of one-dimensional solid-on-solid models of finitesize $L$. For all these lattice models, in the large $L$ limit, a central limitargument shows that, for periodic boundary conditions, $P(h_m,L)$ takes auniversal scaling form $P(h_m,L) \sim (\sqrt{12}w_L)^{-1}f(h_m/(\sqrt{12}w_L))$, with $w_L$ the width of the fluctuating interface and $f(x)$ the Airydistribution function. For one instance of these models, corresponding to theextremely anisotropic Ising model in two dimensions, this result is obtained byan exact computation using transfer matrix technique, valid for any $L>0$.These arguments and exact analytical calculations are supported by numericalsimulations, which show in addition that the subleading scaling function isalso universal, up to a non universal amplitude, and simply given by thederivative of the Airy distribution function $f'(x)$.Comment: 13 pages, 4 figure

Universal asymptotic statistics of maximal relative height in one-dimensional solid-on-solid models