  We report the magnetic-field induced Bose glass$-$BEC transition of magnons
in Tl$_{1-x}$K$_x$CuCl$_3$ and its critical behavior investigated through
specific heat and ESR measurements. The field dependence of the BEC transition
temperature $T_{\rm N}$ can be described by the power law $\left[H\,{-}\,H_{\rm
c}\right]\,{\propto}\,T_{\rm N}^{\phi}$ near the quantum critical point $H_{\rm
c}\,{\sim}\,3.5$ T. The critical exponent ${\phi}$ tends to reach a value
smaller than 1/2 with decreasing fitting window in contrast with
${\phi}\rightarrow 3/2$ for the standard BEC in pure system. At sufficiently
low temperatures, the ESR line shape for $H\,{\simeq}\,H_{\rm c}$ is
intermediate between Gaussian and Lorentzian. This indicates the localization
of magnons for $H\,{<}H_{\rm c}$ at T=0.
