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Lattice-matched InAlN/AlN/GaN high electron mobility transistors offer high performance with attractive electronic and thermal properties. For high-voltage applications, gate leakage currents under reverse bias voltages remain a serious challenge. This current flow is dominated by field enhanced thermal emission from trap states or direct tunneling. We experimentally measure reverse-bias gate leakage currents in InAlN/AlN/GaN transistors at various temperatures and find that the conventional trap-assisted Frenkel-Poole model fails to explain the experimental data. Unlike the non-polar semiconductors Si, Ge, large polarization-induced electric fields exist in III-nitride heterojunctions. When the large polarization fields are accounted for, a modified Frenkel-Poole model is found to accurately explain the measured data at low reverse bias voltages. At high reverse bias voltages, we identify that the direct Fowler-Nordheim tunneling mechanism dominates. The accurate identification of the gate leakage current flow mechanism in these structures leads to the extraction of several useful physical parameters, highlights the importance of polarization fields, and leads to suggestions for improved behavior.