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We study the transience of algebraic varieties in linear groups. Inparticular, we show that a "non elementary" random walk in SL_2(R) escapesexponentially fast from every proper algebraic subvariety. We also treat thecase where the random walk is on the real points of a semi-simple splitalgebraic group and show such a result for a wide family of random walks. As anapplication, we prove that generic subgroups (in some sense) of linear groupsare Zariski dense.

Transience of algebraic varieties in linear groups and application to generic Zariski density