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We consider functions of multi-dimensional versions of truncated Wiener--Hopfoperators with smooth symbols, and study the scaling asymptotics of theirtraces. The obtained results extend the asymptotic formulas obtained by H.Widom in the 1980's to non-smooth functions, and non-smooth truncation domains.The obtained asymptotic formulas are used to analyse the scaling limit of thespatially bipartite entanglement entropy of thermal equilibrium states ofnon-interacting fermions at positive temperature.

On the Szeg\H{o} formulas for truncated Wiener-Hopf operators