We compute the fluctuations of the number of bosons with a given momentum for the Tonks-Girardeau gas at zero and finite temperature in a harmonic trap. We show that correlations between opposite momentum states $p$, which is an important fingerprint of long range order in weakly interacting Bose systems are suppressed. Non trivial correlations, including negative correlations are observed for momenta smaller or of the order of the inverse radius of the gas. The full distribution of the number of bosons with momentum $p$ exhibits an interesting crossover from a non trivial distribution at zero momentum to an exponential distribution. The distribution of the quasi-condensate occupation is also studied. Experimental relevance of our findings for recent cold atoms experiments are discussed.

We compute the fluctuations of the number of bosons with a given momentum for the Tonks-Girardeau gas at zero temperature. We show that correlations between opposite momentum states, which are an important identifying characteristic of long-range order in weakly interacting Bose systems, are suppressed and that the full distribution of the number of bosons with nonzero momentum is exponential. The distribution of the quasicondensate is however quasi-Gaussian. The experimental relevance of our findings for recent cold-atom experiments is discussed.

We compute various current-correlation functions of electrons flowing from a topological nanowire to the tip of a superconducting scanning tunnel microscope and identify fingerprints of a Majorana bound state. In particular, the spin resolved cross correlations are shown to display a clear distinction between the presence of a such an exotic state (negative correlations) and an Andreev bound state (positive correlations). Similarity and differences with measurements with a normal tunnel microscope are also discussed, like the robustness to finite temperature, for instance.

We consider a normal–superconducting junction in order to investigate the effect of new physical ingredients on waiting times. First, we study the interplay between Andreev and specular scattering at the interface on the distribution of waiting times of electrons or holes separately. In that case the distribution is not altered dramatically compared to the case of a single quantum channel with a quantum point contact since the interface acts as an Andreev mirror for holes. We then consider a fully entangled state originating from splitting of Cooper pairs at the interface and demonstrate a significant enhancement of the probability to detect two consecutive electrons in a short time interval. Finally, we discuss the electronic waiting time distribution in the more realistic situation of partial entanglement.

We consider a biased Normal-Superconducting junction with various types of superconductivity. Depending on the class of superconductivity, a Majorana bound state may appear at the interface. We show that this has important consequences on the distribution of waiting times of electrons flowing out of such an interface. Therefore, the waiting time distribution is shown to be a clear fingerprint of Majorana bound state physics and may be considered as an experimental signature of its presence.

We consider a sequence of quantized Lorentzian pulses of noninteracting electrons impinging on a quantum point contact and study the waiting time distribution (WTD), for any transmission and any number of pulses. As the degree of overlap between the electronic wave functions is tuned, the WTD reveals how the correlations between particles are modified. In the weak overlap regime, the WTD is made of several equidistant peaks, separated by the same period as the incoming pulses, contained in an almost exponentially decaying envelope. In the other limit, the WTD of a single quantum channel subjected to a constant voltage is recovered. In both cases, the WTD stresses the difference between the fluctuations induced by the scatterer and the ones encoded in the incoming quantum state. A clear crossover between these two situations is studied with numerical and analytical calculations based on scattering theory.

We consider a N-dot-S junction in the Kondo regime in the limit where the superconducting gap is much smaller than the Kondo temperature. A generalization of the floating of the Kondo resonance is proposed and many body corrections to the average subgap current are calculated. The zero frequency noise is computed and the Fano factor sticks to the value 10/3 for all voltages below the gap. Implications for finite frequency noise are briefly discussed.

We determine by Monte Carlo simulations the width of an interface between the stable phase and the metastable phase in a two-dimensional Ising model with a magnetic field, in the case of nonconversed order parameter (Glauber dynamics). At zero temperature, the width increases ast β withβ−1/3, as predicted by earlier theories. As temperature increases, the value of the effective exponentβ that we measure decreases toward the value 1/4, which is the value in the absence of magnetic field.

In the quantum transport problem of a tight-binding Anderson model, the statistics of eigenvalues for the transfer matrices of thin disordered slabs is studied. Numerical simulations indicate that the probability distribution of nearest neighbor eigenvalue spacing and theΔ3 statistics have already become close to that of the Gaussian orthogonal ensemble for sample lengths of the order of the mean free path, provided that transverse localization effects are not important. An intuitive argument is given why this should occur independently of the size of the matrix. Therefore, good mixing of the channels is not essential for obtaining Gaussian orthogonal ensemble type statistics and universal conductance fluctuations.

An exact enumeration approach is developed for the directed polymer problem. The probability distribution of the number of directed self-avoiding walks that can reach a certain level t is obtained exactly up to t=10. This enables us to calculate some properties of directed polymers that are not attainable by Monte Carlo simulations. Specifically, we find that the fluctuation of the logarithm of the number of directed self-avoiding walks that can reach level t, when averaged over the configurations that can reach level t, scales as ${\mathit{t}}^{1/2}$ well below the directed percolation threshold ${\mathit{p}}_{\mathit{c}\mathit{D}}$, contrary to the behavior ${\mathit{t}}^{1/3}$, which is known to be valid when the bond probability p is above ${\mathit{p}}_{\mathit{c}\mathit{D}}$. When p is close to 1, these fluctuations scale as ${\mathit{t}}^{1/5}$ for a very long time before the true asymptotic behavior ${\mathit{t}}^{1/3}$ is recovered. The method can also be used to obtain the behavior of averages of moments of the number of directed self-avoiding walks that can reach level t. Below ${\mathit{p}}_{\mathit{c}\mathit{D}}$ these quantities are dominated by rare configurations and cannot be obtained by Monte Carlo simulations.

We study the scaling properties of noise reduced Eden clusters in three and four dimensions for variant B in the strip geometry. We find that the width W for large times behaves as a(s)g(Lsd−1), where L is the width of the strip, s the noise reduction parameter, d the dimension of space, and a(s) a decreasing function of s, g is a scaling function with the property g(u)→12 as u→0 and g(u)∼ux as u→∞, where χ is the roughness exponent. This scaling result leads to a new way of determining χ. In 3 dimensions, our numerical values for χ support a recent conjecture by Kim and Kosterlitz: χ = 2(d + 2), and contradict all the former analytical conjectures. In 4 dimensions, we cannot distinguish between the conjectures of Kim and Kosterlitz and the conjecture of Wolf and Kertész, because large crossovers and finite size effects make the measurement of the exponents difficult.

We study the scaling properties of noise reduced Eden clusters in three and four dimensions for variant B in the strip geometry. We find that the width W for large times behaves as a(s)g( L s d−1 ) , where L is the width of the strip, s the noise reduction parameter, d the dimension of space, and a(s) a decreasing function of s , g is a scaling function with the property g(u)→ 1 2 as u →0 and g(u) ∼ u x as u →∞, where χ is the roughness exponent. This scaling result leads to a new way of determining χ. In 3 dimensions, our numerical values for χ support a recent conjecture by Kim and Kosterlitz: χ = 2 (d + 2) , and contradict all the former analytical conjectures. In 4 dimensions, we cannot distinguish between the conjectures of Kim and Kosterlitz and the conjecture of Wolf and Kertész, because large crossovers and finite size effects make the measurement of the exponents difficult.

We present a theoretical study of the localization phenomenon of gravity waves by a rough bottom in a one-dimensional channel. After recalling localization theory and applying it to the shallow-water case, we give the first study of the localization problem in the framework of the full potential theory; in particular we develop a renormalized-transfer-matrix approach to this problem. Our results also yield numerical estimates of the localization length, which we compare with the viscous dissipation length. This allows the prediction of which cases localization should be observable in and in which cases it could be hidden by dissipative mechanisms.

When the interface of a two-dimensional lattice gas is grown by an algorithm producing self-avoiding walks, some features of the Eden model appear. The scaling behavior of the widths and lengths of these interfaces and of their accessible perimeters are examined in several regions of the phase diagram. It is found that both the accessible perimeter and the interface behave like the Eden model below a critical point. Above this point, only the accessible perimeter behaves like the Eden model. The behavior in the critical region is suprisingly the same for the interface and its external perimeter but is different from the Eden model.

We study the evolution of interfaces for a generalization of a diffusion-limited aggregation process in two dimensions, where the walkers are launched from any unoccupied site and have a lifetime $\ensuremath{\tau}$. For $\ensuremath{\tau}=1$, the model reduces to variant $B$ of the Eden model. However, Eden model $B$ does not respond to noise reduction in the same way as the other forms of the Eden model ($A \mathrm{and} C$). As $\ensuremath{\tau}$ increases, stable cellular patterns emerge that resemble the patterns in directional solidification.

Diffusion-limited-aggregation (DLA) deposits with no branching (``zeroth-order branching DLA'') resemble a ``forest of whiskers;'' the fractal dimension ${d}_{f}$ of this forest is significantly less than that of normal DLA deposits. We study a generalization of this whisker model in which only first-order branching is allowed: attachment to the sides as well as the tips of the whiskers is permitted. Using various analysis methods, we find that ${d}_{f}$ may be only very slightly smaller than ${d}_{f}$ for conventional DLA (the full infinite-order branching case).

We present the first evidences for the localization of water waves on a rough bottom. This is achieved through a very precise experimental setting, as well as an extension of localization theory to the full potential theory of hydrodynamics. For the first time the resonant modes due to the disorder are directly observed experimentally. Preliminary nonlinear results are presented.

This is the first study of one of the transmission problems associate to the non-linear Schrödinger equation with a random potential. We show that for almost every realization of the medium the rate of transmission vanishes when increasing the size of the medium; however, whereas it decays exponentially in the linear regime, it decays polynomially in the nonlinear one.