Un der some numerically checkable conditions\, we establish the optimal local law\, i.e.\, we show that the empirical spectral distribution on scales j ust above the eigenvalue spacing follows the global density of states whic h is determined by free probability theory. First\, we give a brief introd uction to the linearization technique that allows to transform the polynom ial model into a linear one\, which has simpler correlation structure but higher dimension. After that we show that the local law holds up to the op timal scale for the generalized resolvent of the linearized model\, which also yields the local law for polynomials. Finally\, we show how the above results can be applied to prove the optimal bulk local law for two concre te families of polynomials: general quadratic forms in Wigner matrices and symmetrized products of independent matrices with i.i.d. entries.This is a joint work with Laszlo Erdö\;s and Torben Krü\;ger .

LOCATION:Big Seminar room Ground floor / Office Bldg West (I21.EG.101)\, IS T Austria ORGANIZER:jdeanton@ist.ac.at SUMMARY:Local laws for polynomials of Wigner matrices URL:https://talks-calendar.app.ist.ac.at/events/1408 END:VEVENT END:VCALENDAR