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Motivated by the thermal transport problem in the Kitaev spin liquids, weconsider a nearest-neighbor tight-binding model on the honeycomb lattice in thepresence of random uncorrelated $\pi$-fluxes. We employ different numericalmethods to study its transport properties near half-filling. Thezero-temperature DC conductivity away from the Dirac point is found to bequadratic in Fermi momentum and inversely proportional to the flux density.Localization due to the random $\pi$-fluxes is observed and the localizationlength is extracted. Our results imply that, for realistic system size, thethermal conductivity of a pure Kitaev spin liquid diverges as$\kappa_\text{K}\sim T^3 e^{\Delta_v/k_BT}$ when $k_B T\ll \Delta_v$, andsuggest the possible occurrence of strong Majorana localization$\kappa_\text{K}/T\ll k_B^2/2\pi\hbar$ when $k_B T\sim \Delta_v$, where$\Delta_v$ is the vison gap.

Transport in honeycomb lattice with random $\pi$-fluxes: implications for low-temperature thermal transport in the Kitaev spin liquids