Constraining Brown Dwarf Desert Formation Mechanisms Through Bayesian Statistical Comparison of Observed and Simulated Populations

We present a comprehensive Bayesian statistical analysis of brown dwarf companions to investigate the physical mechanisms responsible for the observed ``brown dwarf desert'' -- the notable paucity of brown dwarf companions at orbital separations $<$5~AU. Using a carefully vetted sample of 88 confirmed brown dwarf companions from the \texttt{exoplanet.eu} catalog with masses 13--80~$\mjup$ and semi-major axes 0.1--5.0~AU, we employ Markov Chain Monte Carlo (MCMC) optimization and two-dimensional Kolmogorov-Smirnov tests to compare observed orbital and mass distributions with three theoretical formation scenarios: (A) Type II disk-driven migration, (B) core accretion with mass-dependent survival, and (C) dynamical scattering from wide orbits. Our analysis spans 4-parameter models for each scenario, with proper posterior distributions quantifying parameter uncertainties and correlations. The disk migration model provides statistically superior fits (2D KS $p = 0.18$), with optimal parameters $\log_{10}\nu = -6.47^{+0.42}_{-0.31}$, $\sigma_\nu = 0.34^{+0.23}_{-0.17}$, $t_{\rm disk} = 1.66^{+1.24}_{-0.84}$~Myr, and $M_{\rm gap} = 12.0^{+4.7}_{-8.3}~\mjup$, consistent with Type II migration theory. The dynamical scattering model achieves intermediate performance ($p = 0.08$), while core accretion scenarios show poor agreement ($p < 0.001$) despite theoretical sophistication. Occurrence rate analysis reveals the desert region (0.1--5~AU) is depleted by a factor of $\approx$1.6 relative to wide separations ($>$5~AU), a constraint successfully reproduced only by the migration model. Our results provide quantitative evidence that brown dwarfs form at wide separations (10--30~AU) through disk fragmentation and undergo limited Type II migration to reach observed close-in locations, with migration naturally halting near 1~AU through gap-opening processes.