In this paper, the design, modeling, and characterization of a display compatible pMUT platform dedicated for mid air ultrasound gesture recognition is presented. A FEM model has been built using COMSOL for evaluating the frequency response, static profile, acoustic pressure, and driving efficiency of our pMUT device across all vibration modes of circular plates. In parallel with it, a first mode analytical model has been developed including electrical, mechanical, and acoustic domains to provide fast estimation for future design. A laser Doppler vibrometer is used to measure the frequency response, displacement, velocity as well as mode shapes of pMUTs with different designs in the air. Besides, an optical profilometer and impedance meter are used to check the static profile and electrical impedance of devices of different sizes, respectively. Finally, a standard reference microphone is used to measure the acoustic pressure of a pMUT inside its frequency range (<125 kHz). The measured resonance frequency of the first mode across 121.5kHz to 1.16MHz with radius from 500um to 120um fits the prediction of FEM and analytical models. The measurements of the acoustic pressure on the transverse axis of a 500um pMUT also fit the values from the acoustic analytical model.