  Two-dimensional potential flow of the ideal incompressible fluid with free
surface and infinite depth can be described by a conformal map of the fluid
domain into the complex lower half-plane. Stokes wave is the fully nonlinear
gravity wave propagating with the constant velocity. The increase of the scaled
wave height $H/\lambda$ from the linear limit $H/\lambda=0$ to the critical
value $H_{max}/\lambda$ marks the transition from the limit of almost linear
wave to a strongly nonlinear limiting Stokes wave. Here $H$ is the wave height
and $\lambda$ is the wavelength. We simulated fully nonlinear Euler equations,
reformulated in terms of conformal variables, to find Stokes waves for
different wave heights. Analyzing spectra of these solutions we found in
conformal variables, at each Stokes wave height, the distance $v_c$ from the
lowest singularity in the upper half-plane to the real line which corresponds
to the fluid free surface. We also identified that this singularity is the
square-root branch point. The limiting Stokes wave emerges as the singularity
reaches the fluid surface. From the analysis of data for $v_c\to 0$ we suggest
a new power law scaling $v_c\propto (H_{max}-H)^{3/2}$ as well as new estimate
$H_{max}/\lambda \simeq 0.1410633$.
