The non--mesonic weak decay of double--$Lambda$ hypernuclei is studied within a microscopic diagrammatic approach. Besides the nucleon--induced mechanism, $Lambda Nto nN$, widely studied in single--$Lambda$ hypernuclei, additional hyperon--induced me
chanisms, $Lambda Lambdato Lambda n$, $Lambda Lambdato Sigma^0 n$ and $Lambda Lambdato Sigma^-p$, are accessible in double--$Lambda$ hypernuclei and are investigated here. As in previous works on single--$Lambda$ hypernuclei, we adopt a nuclear matter formalism extended to finite nuclei via the local density approximation and a one--meson exchange weak transition potential (including the ground state pseudoscalar and vector octets mesons) supplemented by correlated and uncorrelated two--pion--exchange contributions. The weak decay rates are evaluated for hypernuclei in the region of the experimentally accessible light hypernuclei $^{10}_{LambdaLambda}$Be and $^{13}_{LambdaLambda}$B. Our predictions are compared with a few previous evaluations. The rate for the $Lambda Lambdato Lambda n$ decay is dominated by $K$--, $K^*$-- and $eta$--exchange and turns out to be about 2.5% of the free $Lambda$ decay rate, $Gamma_{Lambda}^{rm free}$, while the total rate for the $Lambda Lambdato Sigma^0 n$ and $Lambda Lambdato Sigma^- p$ decays, dominated by $pi$--exchange, amounts to about 0.25% of $Gamma_{Lambda}^{rm free}$. The experimental measurement of these decays would be essential for the beginning of a systematic study of the non--mesonic decay of strangeness $-2$ hypernuclei. This field of research could also shed light on the possible existence and nature of the $H$--dibaryon.
