  We study the effect of thermal fluctuations on the wetting phase transitions
of infinite order and of continuously varying order, recently discovered within
a mean-field density-functional model for three-phase equilibria in systems
with short-range forces and a two-component order parameter. Using linear
functional renormalization group (RG) calculations within a local interface
Hamiltonian approach, we show that the infinite-order transitions are robust.
The exponential singularity (implying $2-\alpha_s = \infty$) of the surface
free energy excess at infinite-order wetting as well as the precise algebraic
divergence (with $\beta_s = -1$) of the wetting layer thickness are not
modified as long as $\omega < 2$, with $\omega$ the dimensionless wetting
parameter that measures the strength of thermal fluctuations. The interface
width diverges algebraically and universally (with $\nu_{\perp} = 1/2$). In
contrast, the non-universal critical wetting transitions of finite but
continuously varying order are modified when thermal fluctuations are taken
into account, in line with predictions from earlier calculations on similar
models displaying weak, intermediate and strong fluctuation regimes.
