Particle Transport within the Pulsar Wind Nebula HESS J1825-137

Aims: We present a detailed view of the pulsar wind nebula (PWN) HESS J1825-137. We aim to constrain the mechanisms dominating the particle transport within the nebula, accounting for its anomalously large size and spectral characteristics. Methods: The nebula is studied using a deep exposure from over 12 years of H.E.S.S. I operation, together with data from H.E.S.S. II improving the low energy sensitivity. Enhanced energy-dependent morphological and spatially-resolved spectral analyses probe the Very High Energy (VHE, E > 0.1 TeV) gamma-ray properties of the nebula. Results: The nebula emission is revealed to extend out to 1.5 degrees from the pulsar, ~1.5 times further than previously seen, making HESS J1825--137, with an intrinsic diameter of ~100 pc, potentially the largest gamma-ray PWN currently known. Characterisation of the nebula's strongly energy-dependent morphology enables the particle transport mechanisms to be constrained. A dependence of the nebula extent with energy of R $\propto$ E^αwith α= -0.29 +/- 0.04 (stat) +/- 0.05 (sys) disfavours a pure diffusion scenario for particle transport within the nebula. The total gamma-ray flux of the nebula above 1~TeV is found to be (1.12 +/- 0.03 (stat) +/- 0.25 (sys)) $\times 10^{-11}$ cm$^{-2}$ s$^{-1}$, corresponding to ~64% of the flux of the Crab Nebula. Conclusions: HESS J1825-137 is a PWN with clear energy-dependent morphology at VHE gamma-ray energies. This source is used as a laboratory to investigate particle transport within middle-aged PWNe. Deep observations of this highly spatially-extended PWN enable a spectral map of the region to be produced, providing insights into the spectral variation within the nebula.