Dynamical Lifshitz Tails
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866910264091213824 |
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| author | Emilsdóttir, Íris Monakov, Grigorii |
| author_facet | Emilsdóttir, Íris Monakov, Grigorii |
| contents | We consider one-parameter families of random circle diffeomorphisms $g_{E,y}$ for which the unperturbed map $g_{0,\bar{0}}$ has a fixed point of order $2k$ and the dependence on the parameter $E$ is monotone. Under reasonable assumptions, we show that the rotation number $ρ(E)$ exhibits Lifshitz tail decay with exponent $-\frac{2k - 1}{2k}$, \[ \lim_{E \downarrow 0} \frac{\ln(-\ln(ρ(E) - ρ(0)))}{\ln(E)} = -\frac{2k-1}{2k}. \] The exponent is determined by the passage time through a parabolic bottleneck. A full rotation requires on the order of $E^{-\frac{2k - 1}{2k}}$ successive small perturbations, and the probability of such a streak decays exponentially as a function of its length. When $k=1$, the exponent is $-1/2$, and we recover as a corollary a purely dynamical proof of Lifshitz tail asymptotics at the spectral edges of the one-dimensional Anderson model. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_27793 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Dynamical Lifshitz Tails Emilsdóttir, Íris Monakov, Grigorii Dynamical Systems Spectral Theory 37E45, 37E10, 37H99, 47B80 We consider one-parameter families of random circle diffeomorphisms $g_{E,y}$ for which the unperturbed map $g_{0,\bar{0}}$ has a fixed point of order $2k$ and the dependence on the parameter $E$ is monotone. Under reasonable assumptions, we show that the rotation number $ρ(E)$ exhibits Lifshitz tail decay with exponent $-\frac{2k - 1}{2k}$, \[ \lim_{E \downarrow 0} \frac{\ln(-\ln(ρ(E) - ρ(0)))}{\ln(E)} = -\frac{2k-1}{2k}. \] The exponent is determined by the passage time through a parabolic bottleneck. A full rotation requires on the order of $E^{-\frac{2k - 1}{2k}}$ successive small perturbations, and the probability of such a streak decays exponentially as a function of its length. When $k=1$, the exponent is $-1/2$, and we recover as a corollary a purely dynamical proof of Lifshitz tail asymptotics at the spectral edges of the one-dimensional Anderson model. |
| title | Dynamical Lifshitz Tails |
| topic | Dynamical Systems Spectral Theory 37E45, 37E10, 37H99, 47B80 |
| url | https://arxiv.org/abs/2605.27793 |