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- W2276407217 abstract "We consider the Anderson polymer partition function $$ u(t):=mathbb{E}^XBigl[e^{int_0^t mathrm{d}B^{X(s)}_s}Bigr],, $$ where ${B^{x}_t,;, tgeq0}_{xinmathds{Z}^d}$ is a family of independent fractional Brownian motions all with Hurst parameter $Hin(0,1)$, and ${X(t)}_{tin mathds{R}^{+}}$ is a continuous-time simple symmetric random walk on $mathds{Z}^d$ with jump rate $kappa$ and started from the origin. $mathbb{E}^X$ is the expectation with respect to this random walk. We prove that when $H$ is less than a half $u(t)$ grows exponentially in $t$. More precisely, we show that ${frac{1}{t}log u(t)}_{tin mathds{R}^{+}}$ converges both in $mathcal{L}^1$ and almost surely to some deterministic positive number for every $kappainmathds{R}^+$. For $H>1/2$, we show that almost surely $log u(t)$ grows at least linearly in $t$, and at most as $c tsqrt{log t}$ for some deterministic positive constant $c$. The same is proved for its expectation. Finally, we show that when $mathds{Z}^d$ is replaced by a compact space with a Holder continuous covariance structure, the above-mentioned upper-bound $c tsqrt{log t}$ can be improved to $c_1 t$." @default.
- W2276407217 created "2016-06-24" @default.
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- W2276407217 date "2016-02-17" @default.
- W2276407217 modified "2023-09-27" @default.
- W2276407217 title "Asymptotic behavior of the Anderson polymer in a fractional Brownian environment" @default.
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