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Register a new userSpectral estimates for Ruelle operators with two parameters and sharp large deviations
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We obtain spectral estimates for the iterations of Ruelle operator $L_{f + (a + i b)tau + (c + i d) g}$ with two complex parameters and H{o}lder functions $f,: g$ generalizing the case $Pr(f) =0$ studied in [PeS2]. As an application we prove a sharp large deviation theorem concerning exponentially shrinking intervals which improves the result in [PeS1].
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For hyperbolic flows $varphi_t$ we examine the Gibbs measure of points $w$ for which $$int_0^T G(varphi_t w) dt - a T in (- e^{-epsilon n}, e^{- epsilon n})$$ as $n to infty$ and $T geq n$, provided $epsilon > 0$ is sufficiently small. This is similar to local central limit theorems. The fact that the interval $(- e^{-epsilon n}, e^{- epsilon n})$ is exponentially shrinking as $n to infty$ leads to several difficulties. Under some geometric assumptions we establish a sharp large deviation result with leading term $C(a) epsilon_n e^{gamma(a) T}$ and rate function $gamma(a) leq 0.$ The proof is based on the spectral estimates for the iterations of the Ruelle operators with two complex parameters and on a new Tauberian theorem for sequence of functions $g_n(t)$ having an asymptotic as $ n to infty$ and $t geq n.$
We consider the Dirichlet Laplace operator on open, quasi-bounded domains of infinite volume. For such domains semiclassical spectral estimates based on the phase-space volume - and therefore on the volume of the domain - must fail. Here we present a method how one can nevertheless prove uniform bounds on eigenvalues and eigenvalue means which are sharp in the semiclassical limit. We give examples in horn-shaped regions and so-called spiny urchins. Some results are extended to Schrodinger operators defined on quasi-bounded domains with Dirichlet boundary conditions.
Consider first passage percolation with identical and independent weight distributions and first passage time ${rm T}$. In this paper, we study the upper tail large deviations $mathbb{P}({rm T}(0,nx)>n(mu+xi))$, for $xi>0$ and $x eq 0$ with a time constant $mu$ and a dimension $d$, for weights that satisfy a tail assumption $ beta_1exp{(-alpha t^r)}leq mathbb P(tau_e>t)leq beta_2exp{(-alpha t^r)}.$ When $rleq 1$ (this includes the well-known Eden growth model), we show that the upper tail large deviation decays as $exp{(-(2dxi +o(1))n)}$. When $1< rleq d$, we find that the rate function can be naturally described by a variational formula, called the discrete p-Capacity, and we study its asymptotics. For $r<d$, we show that the large deviation event ${rm T}(0,nx)>n(mu+xi)$ is described by a localization of high weights around the origin. The picture changes for $rgeq d$ where the configuration is not anymore localized.
For non-critical almost Mathieu operators with Diophantine frequency, we establish exponential asymptotics on the size of spectral gaps, and show that the spectrum is homogeneous. We also prove the homogeneity of the spectrum for Schodinger operators with (measure-theoretically) typical quasi-periodic analytic potentials and fixed strong Diophantine frequency. As applications, we show the discrete version of Deifts conjecture cite{Deift, Deift17} for subcritical analytic quasi-periodic initial data and solve a series of open problems of Damanik-Goldstein et al cite{BDGL, DGL1, dgsv, Go} and Kotani cite{Kot97}.
For diffusive many-particle systems such as the SSEP (symmetric simple exclusion process) or independent particles coupled with reservoirs at the boundaries, we analyze the density fluctuations conditioned on current integrated over a large time. We determine the conditioned large deviation function of density by a microscopic calculation. We then show that it can be expressed in terms of the solutions of Hamilton-Jacobi equations, which can be written for general diffusive systems using a fluctuating hydrodynamics description.
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