Same entry in Hesychius pi1911 (cf. 1912-1913), the Synagoge, and Photius, Lexicon pi784 Theodoridis.
The headword is classed in modern dictionaries as the third person singular of the future indicative of peritugxa/nw, always in the middle (but see tau 430, tau 435). Of the two definitions the first can only be an aorist subjunctive active, suggesting that the headword is a desiderative future (tau 430). Both definitions are from verbs in which the third person singular is often an impersonal "it happens" (sigma 1639, web address 1). The headword is never, I believe, used as an impersonal. See LSJ (web address 2).
For the controversial question of the degree of chance implied in words related to the verb e)/tuxon and to its antonyms a)pe/tuxon and the aorists of a(marta/nw, verbs of hitting or missing a target in Homer, see epsilon 3344, eta 286, alpha 3591 and delta 641 (cf. tau 1234 on tu/xh), with their cross-references and references.