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\textbf{SPEAKER}:
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%\textbf{SPEAKER}:
Fr\'ed\'eric Klopp,  Sorbonne Universit\'e and UC Berkeley
 
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{\bf TITLE}:
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On the spatial extent of localized eigenfunctions for
        random Schr\"dinger operators
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{\bf ABSTRACT:}\\
On $\mathbb{Z}^d$, consider $\varphi$, an
       $\ell^2-$normalized function that decays exponentially at
        $\infty$ at a rate at least $\mu$. One can define the
        \emph{onset length} (of the exponential decay) of $\varphi$ as
        the radius of the smallest ball, say, $B$, such that one has
        the following global bound $|\varphi(x)|\leq
        \|\varphi\|_\infty e^{-\mu\,\text{dist}(x,B)}$.
        The present talk will describe the onset lengths of the
        localized eigenfunctions of random Schrödinger operators.
        Under suitable assumptions, we prove that, with probability
        one, the number of eigenfunctions in the localization regime
        having onset length larger than $\ell$ and localization center
        in a ball of radius $L$ is smaller than $C L^d\exp(-c \ell)$,
        for $\ell>0$ large (for some constants $C,c>0$). Thus, most
        eigenfunctions localize on small size balls independent of the
        system size which is the physicists understanding of
        localization; to our knowledge, this did not result from
        existing mathematical estimates. The talk is based on joint
        work with Jeff Schenker.



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