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The Gaussian entropy map in valued fields

### Yassine El Maazouz

Vol. 13 (2022), No. 1, 1–18
##### Abstract

The entropy map for multivariate real valued Gaussian distributions is the map that sends a positive definite matrix $\mathrm{\Sigma }$ to the sequence of logarithms of its principal minors ${\left(\mathrm{log}\left(\mathrm{det}{\mathrm{\Sigma }}_{I}\right)\right)}_{I}$. We exhibit the analog of this map in the nonarchimedean local fields setting, like the field of $p$-adic numbers for example. As in the real case, the image of this map lies in the supermodular cone. Moreover, given a multivariate Gaussian measure on a local field, its image under the entropy map determines its pushforward under valuation. In general, this map can be defined for nonarchimedean valued fields whose valuation group is an additive subgroup of the real line, and it remains supermodular. We also explicitly compute the image of this map in dimension $3$.

##### Keywords
entropy, probability, Gaussian measures, nonarchimedean valuation, local fields, Bruhat–Tits building, conditional independence
##### Mathematical Subject Classification
Primary: 12J25, 60E05, 94A17