  We propose a second-order statistic parameter $\varepsilon$, the relative
occurrence rate between hot and cold Jupiters ($\varepsilon=\eta_{\rm
HJ}/\eta_{\rm CJ}$), to probe the migration of gas giants. Since the planet
occurrence rate is the combined outcome of the formation and migration
processes, a joint analysis of hot and cold Jupiter frequency may shed light on
the dynamical evolution of giant planet systems. We first investigate the
behavior of $\varepsilon$ as the stellar mass changes observationally. Based on
the occurrence rate measurements of hot Jupiters ($\eta_{\rm HJ}$) from the
TESS survey and cold Jupiters ($\eta_{\rm CJ}$) from the CLS survey, we find a
tentative trend (97% confidence) that $\varepsilon$ drops when the stellar mass
rises from $0.8$ to $1.4\ M_\odot$, which can be explained by different giant
planet growth and disk migration timescales around different stars. We carry
out planetesimal and pebble accretion simulations, both of which could
reproduce the results of $\eta_{\rm HJ}$, $\eta_{\rm CJ}$ and $\varepsilon$.
Our findings indicate that the classical core accretion + disk migration model
can explain the observed decreasing trend of $\varepsilon$. We propose two ways
to increase the significance of the trend and verify the anti-correlation.
Future works are required to better constrain $\varepsilon$, especially for M
dwarfs and for more massive stars.
