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Register a new userTruncated linear statistics associated with the eigenvalues of random matrices II. Partial sums over proper time delays for chaotic quantum dots
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Invariant ensembles of random matrices are characterized by the distribution of their eigenvalues ${lambda_1,cdots,lambda_N}$. We study the distribution of truncated linear statistics of the form $tilde{L}=sum_{i=1}^p f(lambda_i)$ with $p<N$. This problem has been considered by us in a previous paper when the $p$ eigenvalues are further constrained to be the largest ones (or the smallest). In this second paper we consider the same problem without this restriction which leads to a rather different analysis. We introduce a new ensemble which is related, but not equivalent, to the thinned ensembles introduced by Bohigas and Pato. This question is motivated by the study of partial sums of proper time delays in chaotic quantum dots, which are characteristic times of the scattering process. Using the Coulomb gas technique, we derive the large deviation function for $tilde{L}$. Large deviations of linear statistics $L=sum_{i=1}^N f(lambda_i)$ are usually dominated by the energy of the Coulomb gas, which scales as $sim N^2$, implying that the relative fluctuations are of order $1/N$. For the truncated linear statistics considered here, there is a whole region (including the typical fluctuations region), where the energy of the Coulomb gas is frozen and the large deviation function is purely controlled by an entropic effect. Because the entropy scales as $sim N$, the relative fluctuations are of order $1/sqrt{N}$. Our analysis relies on the mapping on a problem of $p$ fictitious non-interacting fermions in $N$ energy levels, which can exhibit both positive and negative effective (absolute) temperatures. We determine the large deviation function characterizing the distribution of the truncated linear statistics, and show that, for the case considered here ($f(lambda)=1/lambda$), the corresponding phase diagram is separated in three different phases.
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An invariant ensemble of $Ntimes N$ random matrices can be characterised by a joint distribution for eigenvalues $P(lambda_1,cdots,lambda_N)$. The study of the distribution of linear statistics, i.e. of quantities of the form $L=(1/N)sum_if(lambda_i)$ where $f(x)$ is a given function, appears in many physical problems. In the $Ntoinfty$ limit, $L$ scales as $Lsim N^eta$, where the scaling exponent $eta$ depends on the ensemble and the function $f$. Its distribution can be written under the form $P_N(s=N^{-eta},L)simeq A_{beta,N}(s),expbig{-(beta N^2/2),Phi(s)big}$, where $betain{1,,2,,4}$ is the Dyson index. The Coulomb gas technique naturally provides the large deviation function $Phi(s)$, which can be efficiently obtained thanks to a thermodynamic identity introduced earlier. We conjecture the pre-exponential function $A_{beta,N}(s)$. We check our conjecture on several well controlled cases within the Laguerre and the Jacobi ensembles. Then we apply our main result to a situation where the large deviation function has no minimum (and $L$ has infinite moments)~: this arises in the statistical analysis of the Wigner time delay for semi-infinite multichannel disordered wires (Laguerre ensemble). The statistical analysis of the Wigner time delay then crucially depends on the pre-exponential function $A_{beta,N}(s)$, which ensures the decay of the distribution for large argument.
We systematically study the first three terms in the asymptotic expansions of the moments of the transmission eigenvalues and proper delay times as the number of quantum channels n in the leads goes to infinity. The computations are based on the assumption that the Landauer-Butticker scattering matrix for chaotic ballistic cavities can be modelled by the circular ensembles of Random Matrix Theory (RMT). The starting points are the finite-n formulae that we recently discovered (Mezzadri and Simm, J. Math. Phys. 52 (2011), 103511). Our analysis includes all the symmetry classes beta=1,2,4; in addition, it applies to the transmission eigenvalues of Andreev billiards, whose symmetry classes were classified by Zirnbauer (J. Math. Phys. 37 (1996), 4986-5018) and Altland and Zirnbauer (Phys. Rev. B. 55 (1997), 1142-1161). Where applicable, our results are in complete agreement with the semiclassical theory of mesoscopic systems developed by Berkolaiko et al. (J. Phys. A.: Math. Theor. 41 (2008), 365102) and Berkolaiko and Kuipers (J. Phys. A: Math. Theor. 43 (2010), 035101 and New J. Phys. 13 (2011), 063020). Our approach also applies to the Selberg-like integrals. We calculate the first two terms in their asymptotic expansion explicitly.
In this paper, we study the probability distribution of the observable $s = (1/N)sum_{i=N-N+1}^N x_i$, with $1 leq N leq N$ and $x_1<x_2<cdots< x_N$ representing the ordered positions of $N$ particles in a $1d$ one-component plasma, i.e., $N$ harmonically confined charges on a line, with pairwise repulsive $1d$ Coulomb interaction $|x_i-x_j|$. This observable represents an example of a truncated linear statistics -- here the center of mass of the $N = kappa , N$ (with $0 < kappa leq 1$) rightmost particles. It interpolates between the position of the rightmost particle (in the limit $kappa to 0$) and the full center of mass (in the limit $kappa to 1$). We show that, for large $N$, $s$ fluctuates around its mean $langle s rangle$ and the typical fluctuations are Gaussian, of width $O(N^{-3/2})$. The atypical large fluctuations of $s$, for fixed $kappa$, are instead described by a large deviation form ${cal P}_{N, kappa}(s)simeq exp{left[-N^3 phi_kappa(s)right]}$, where the rate function $phi_kappa(s)$ is computed analytically. We show that $phi_{kappa}(s)$ takes different functional forms in five distinct regions in the $(kappa,s)$ plane separated by phase boundaries, thus leading to a rich phase diagram in the $(kappa,s)$ plane. Across all the phase boundaries the rate function $phi(kappa,s)$ undergoes a third-order phase transition. This rate function is also evaluated numerically using a sophisticated importance sampling method, and we find a perfect agreement with our analytical predictions.
We study the spectral statistics of spatially-extended many-body quantum systems with on-site Abelian symmetries or local constraints, focusing primarily on those with conserved dipole and higher moments. In the limit of large local Hilbert space dimension, we find that the spectral form factor $K(t)$ of Floquet random circuits can be mapped exactly to a classical Markov circuit, and, at late times, is related to the partition function of a frustration-free Rokhsar-Kivelson (RK) type Hamiltonian. Through this mapping, we show that the inverse of the spectral gap of the RK-Hamiltonian lower bounds the Thouless time $t_{mathrm{Th}}$ of the underlying circuit. For systems with conserved higher moments, we derive a field theory for the corresponding RK-Hamiltonian by proposing a generalized height field representation for the Hilbert space of the effective spin chain. Using the field theory formulation, we obtain the dispersion of the low-lying excitations of the RK-Hamiltonian in the continuum limit, which allows us to extract $t_{mathrm{Th}}$. In particular, we analytically argue that in a system of length $L$ that conserves the $m^{th}$ multipole moment, $t_{mathrm{Th}}$ scales subdiffusively as $L^{2(m+1)}$. We also show that our formalism directly generalizes to higher dimensional circuits, and that in systems that conserve any component of the $m^{th}$ multipole moment, $t_{mathrm{Th}}$ has the same scaling with the linear size of the system. Our work therefore provides a general approach for studying spectral statistics in constrained many-body chaotic systems.
We study non-equilibrium steady state transport in scale invariant quantum junctions with focus on the particle and heat fluctuations captured by the two-point current correlation functions. We show that the non-linear behavior of the particle current affects both the particle and heat noise. The existence of domains of enhancement and reduction of the noise power with respect to the linear regime are observed. The impact of the statistics is explored. We demonstrate that in the scale invariant case the bosonic particle noise exceeds the fermionic one in the common domain of heat bath parameters. Multi-lead configurations are also investigated and the effect of probe terminals on the noise is discussed.
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