We model the distribution of the observed profiles of 21 cm line emission from neutral hydrogen (HI) in central galaxies selected from a statistically representative mock catalog of the local Universe in the Lambda-cold dark matter framework. The distribution of these HI velocity profiles (specifically, their widths $W_{50}$) has been observationally constrained, but has not been systematically studied theoretically. Our model profiles derive from rotation curves of realistically baryonified haloes in an N-body simulation, including the quasi-adiabatic relaxation of the dark matter profile of each halo in response to its baryons. We study the predicted $W_{50}$ distribution using a realistic pipeline applied to noisy profiles extracted from our luminosity-complete mock catalog with an ALFALFA-like survey geometry and redshift selection. Our default mock is in good agreement with observed ALFALFA results for $W_{50}gtrsim700$ km/s, being incomplete at lower widths due to the intrinsic threshold of $M_rleq-19$. Variations around the default model show that the velocity width function at $W_{50}gtrsim300$ km/s is most sensitive to a possible correlation between galaxy inclination and host concentration, followed by the physics of quasi-adiabatic relaxation. We also study the excess kurtosis of noiseless velocity profiles, obtaining a distribution which tightly correlates with $W_{50}$, with a shape and scatter that depend on the properties of the turbulent HI disk. Our results open the door towards using the shapes of HI velocity profiles as a novel statistical probe of the baryon-dark matter connection.
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