  Quasicritical exponents of one-dimensional models displaying a
quasitransition at finite temperatures are examined in detail. The
quasitransition is characterized by intense sharp peaks in physical quantities
such as specific heat and magnetic susceptibility, which are reminiscent of
divergences accompanying a continuous (second-order) phase transition. The
question whether these robust finite peaks follow some power law around the
quasicritical temperature is addressed. Although there is no actual divergence
of these quantities at a quasicritical temperature, a power-law behavior fits
precisely both ascending as well as descending part of the peaks in the
vicinity but not too close to a quasicritical temperature. The specific values
of the quasicritical exponents are rigorously calculated for a class of
one-dimensional models (e.g. Ising-XYZ diamond chain, coupled spin-electron
double-tetrahedral chain, Ising-XXZ two-leg ladder, and Ising-XXZ three-leg
tube), whereas the same set of quasicritical exponents implies a certain
`universality' of quasitransitions of one-dimensional models. Specifically, the
values of the quasicritical exponents for one-dimensional models are:
$\alpha=\alpha'=3$ for the specific heat, $\gamma=\gamma'=3$ for the
susceptibility and $\nu=\nu'=1$ for the correlation length.
