arXiv subjects

Gregory Schehr

Publications and source records attributed to Gregory Schehr.

At least 109 records · Page 6Linked to original sources

arXiv2019

Intermediate deviation regime for the full eigenvalue statistics in the complex Ginibre ensemble

We study the Ginibre ensemble of $N \times N$ complex random matrices and compute exactly, for any finite $N$, the full distribution as well as all the cumulants of the number $N_r$ of eigenvalues within a disk of radius $r$ centered at the origin. In the limit of large $N$, when the average density of eigenvalues becomes uniform over the unit disk, we show that for $0<r<1$ the fluctuations of $N_r$ around its mean value $\langle N_r \rangle \approx N r^2$ display three different regimes: (i) a typical Gaussian regime where the fluctuations are of order ${\cal O}(N^{1/4})$, (ii) an intermediate regime where $N_r - \langle N_r \rangle = {\cal O}(\sqrt{N})$, and (iii) a large deviation regime where $N_r - \langle N_r \rangle = {\cal O}({N})$. This intermediate behaviour (ii) had been overlooked in previous studies and we show here that it ensures a smooth matching between the typical and the large deviation regimes. In addition, we demonstrate that this intermediate regime controls all the (centred) cumulants of $N_r$, which are all of order ${\cal O}(\sqrt{N})$, and we compute them explicitly. Our analytical results are corroborated by precise "importance sampling" Monte Carlo simulations.

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arXiv2019

Non-crossing run-and-tumble particles on a line

We study active particles performing independent run and tumble motion on an infinite line with velocities $v_0 σ(t)$, where $σ(t) = \pm 1$ is a dichotomous telegraphic noise with constant flipping rate $γ$. We first consider one particle in the presence of an absorbing wall at $x=0$ and calculate the probability that it has survived up to time $t$ and is at position $x$ at time $t$. We then consider two particles with independent telegraphic noises and compute exactly the probability that they do not cross up to time $t$. Contrarily to the case of passive (Brownian) particles this two-RTP problem can not be reduced to a single RTP with an absorbing wall. Nevertheless, we are able to compute exactly the probability of no-crossing of two independent RTP's up to time $t$ and find that it decays at large time as $t^{-1/2}$ with an amplitude that depends on the initial condition. The latter allows to define an effective length scale, analogous to the so called `` Milne extrapolation length'' in neutron scattering, which we demonstrate to be a fingerprint of the active dynamics.

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arXiv2019

Nonequilibrium dynamics of noninteracting fermions in a trap

We consider the real time dynamics of $N$ noninteracting fermions in $d=1$. They evolve in a trapping potential $V(x)$, starting from the equilibrium state in a potential $V_0(x)$. We study the time evolution of the Wigner function $W(x,p,t)$ in the phase space $(x,p)$, and the associated kernel which encodes all correlation functions. At $t=0$ the Wigner function for large $N$ is uniform in phase space inside the Fermi volume, and vanishes at the Fermi surf over a scale $e_N$ being described by a universal scaling function related to the Airy function. We obtain exact solutions for the Wigner function, the density, and the correlations in the case of harmonic and inverse square potentials, for several $V_0(x)$. In the large $N$ limit, near the edges where the density vanishes, we obtain limiting kernels (of the Airy or Bessel types) that retain the form found in equilibrium, up to a time dependent rescaling. For non-harmonic traps the evolution of the Fermi volume is more complex. Nevertheless we show that, for intermediate times, the Fermi surf is still described by the same equilibrium scaling function, with a non-trivial time and space dependent width which we compute analytically. We discuss the multi-time correlations and obtain their explicit scaling forms valid near the edge for the harmonic oscillator. Finally, we address the large time limit where relaxation to the Generalized Gibbs Ensemble (GGE) was found to occur in the "classical" regime $\hbar \sim 1/N$. Using the diagonal ensemble we compute the Wigner function in the quantum case (large $N$, fixed $\hbar$) and show that it agrees with the GGE. We also obtain the higher order (non-local) correlations in the diagonal ensemble.

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arXiv2019

Extremes of $2d$ Coulomb gas: universal intermediate deviation regime

In this paper, we study the extreme statistics in the complex Ginibre ensemble of $N \times N$ random matrices with complex Gaussian entries, but with no other symmetries. All the $N$ eigenvalues are complex random variables and their joint distribution can be interpreted as a $2d$ Coulomb gas with a logarithmic repulsion between any pair of particles and in presence of a confining harmonic potential $v(r) \propto r^2$. We study the statistics of the eigenvalue with the largest modulus $r_{\max}$ in the complex plane. The typical and large fluctuations of $r_{\max}$ around its mean had been studied before, and they match smoothly to the right of the mean. However, it remained a puzzle to understand why the large and typical fluctuations to the left of the mean did not match. In this paper, we show that there is indeed an intermediate fluctuation regime that interpolates smoothly between the large and the typical fluctuations to the left of the mean. Moreover, we compute explicitly this "intermediate deviation function" (IDF) and show that it is universal, i.e. independent of the confining potential $v(r)$ as long as it is spherically symmetric and increases faster than $\ln r^2$ for large $r$ with an unbounded support. If the confining potential $v(r)$ has a finite support, i.e. becomes infinite beyond a finite radius, we show via explicit computation that the corresponding IDF is different. Interestingly, in the borderline case where the confining potential grows very slowly as $v(r) \sim \ln r^2$ for $r \gg 1$ with an unbounded support, the intermediate regime disappears and there is a smooth matching between the central part and the left large deviation regime.

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arXiv2018

Noninteracting fermions in a trap and random matrix theory

We review recent advances in the theory of trapped fermions using techniques borrowed from random matrix theory (RMT) and, more generally, from the theory of determinantal point processes. In the presence of a trap, and in the limit of a large number of fermions $N \gg 1$, the spatial density exhibits an edge, beyond which it vanishes. While the spatial correlations far from the edge, i.e. close to the center of the trap, are well described by standard many-body techniques, such as the local density approximation (LDA), these methods fail to describe the fluctuations close to the edge of the Fermi gas, where the density is very small and the fluctuations are thus enhanced. It turns out that RMT and determinantal point processes offer a powerful toolbox to study these edge properties in great detail. Here we discuss the principal edge universality classes, that have been recently identified using these modern tools. In dimension $d=1$ and at zero temperature $T=0$, these universality classes are in one-to-one correspondence with the standard universality classes found in the classical unitary random matrix ensembles: soft edge (described by the "Airy kernel") and hard edge (described by the "Bessel kernel") universality classes. We further discuss extensions of these results to higher dimensions $d\geq 2$ and to finite temperature. Finally, we discuss correlations in the phase space, i.e., in the space of positions and momenta, characterized by the so called Wigner function.

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arXiv2018

Kinetic energy of a trapped Fermi gas at finite temperature

We study the statistics of the kinetic (or equivalently potential) energy for $N$ non-interacting fermions in a $1d$ harmonic trap of frequency $ω$, at finite temperature $T$. Remarkably, we find an exact solution for the full distribution of the kinetic energy, at any temperature $T$ and for any $N$, using a non-trivial mapping to an integrable Calogero-Moser-Sutherland model. As a function of temperature $T$, and for large $N$, we identify: (i) a quantum regime, for $T \sim \hbar ω$, where quantum fluctuations dominate and (ii) a thermal regime, for $T \sim N \hbar ω$, governed by thermal fluctuations. We show how the mean, the variance as well as the large deviation function associated with the distribution of the kinetic energy cross over from the quantum to the thermal regime as temperature increases.

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arXiv2018

Entanglement Entropy and Full Counting Statistics for $2d$-Rotating Trapped Fermions

We consider $N$ non-interacting fermions in a $2d$ harmonic potential of trapping frequency $ω$ and in a rotating frame at angular frequency $Ω$, with $0<ω- Ω\ll ω$. At zero temperature, the fermions are in the non-degenerate lowest Landau level and their positions are in one to one correspondence with the eigenvalues of an $N\times N$ complex Ginibre matrix. For large $N$, the fermion density is uniform over the disk of radius $\sqrt{N}$ centered at the origin and vanishes outside this disk. We compute exactly, for any finite $N$, the Rényi entanglement entropy of order $q$, $S_q(N,r)$, as well as the cumulants of order $p$, $\langle{N_r^{p}}\rangle_c$, of the number of fermions $N_r$ in a disk of radius $r$ centered at the origin. For $N \gg 1$, in the (extended) bulk, i.e., for $0 < r/\sqrt{N} < 1$, we show that $S_q(N,r)$ is proportional to the number variance ${\rm Var}\,(N_r)$, despite the non-Gaussian fluctuations of $N_r$. This relation breaks down at the edge of the fermion density, for $r \approx \sqrt{N}$, where we show analytically that $S_q(N,r)$ and ${\rm Var}\,(N_r)$ have a different $r$-dependence.

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arXiv2018

Exact persistence exponent for the $2d$-diffusion equation and related Kac polynomials

We compute the persistence for the $2d$-diffusion equation with random initial condition, i.e., the probability $p_0(t)$ that the diffusion field, at a given point ${\bf x}$ in the plane, has not changed sign up to time $t$. For large $t$, we show that $p_0(t) \sim t^{-θ(2)}$ with $θ(2) = 3/16$. Using the connection between the $2d$-diffusion equation and Kac random polynomials, we show that the probability $q_0(n)$ that Kac polynomials, of (even) degree $n$, have no real root decays, for large $n$, as $q_0(n) \sim n^{-3/4}$. We obtain this result by using yet another connection with the truncated orthogonal ensemble of random matrices. This allows us to compute various properties of the zero-crossings of the diffusing field, equivalently of the real roots of Kac polynomials. Finally, we unveil a precise connection with a fourth model: the semi-infinite Ising spin chain with Glauber dynamics at zero temperature.

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arXiv2018

Non-interacting fermions in hard-edge potentials

We consider the spatial quantum and thermal fluctuations of non-interacting Fermi gases of $N$ particles confined in $d$-dimensional non-smooth potentials. We first present a thorough study of the spherically symmetric pure hard-box potential, with vanishing potential inside the box, both at $T=0$ and $T>0$. We find that the correlations near the wall are described by a "hard edge" kernel, which depend both on $d$ and $T$, and which is different from the "soft edge" Airy kernel, and its higher $d$ generalizations, found for smooth potentials. We extend these results to the case where the potential is non-uniform inside the box, and find that there exists a family of kernels which interpolate between the above "hard edge" kernel and the "soft edge" kernels. Finally, we consider one-dimensional singular potentials of the form $V(x)\sim |x|^{-γ}$ with $γ>0$. We show that the correlations close to the singularity at $x=0$ are described by this "hard edge" kernel for $1\leqγ<2$ while they are described by a broader family of "hard edge" kernels known as the Bessel kernel for $γ=2$ and, finally by the Airy kernel for $γ>2$. These one-dimensional kernels also appear in random matrix theory, and we provide here the mapping between the $1d$ fermion models and the corresponding random matrix ensembles. Part of these results were announced in a recent Letter, EPL 120, 10006 (2017).

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arXiv2018

Asymptotics for the Expected Maximum of Random Walks and Lévy Flights with a Constant Drift

In this paper, we study the large $n$ asymptotics of the expected maximum of an $n$-step random walk/Lévy flight (characterized by a Lévy index $1<μ\leq 2$) on a line, in the presence of a constant drift $c$. For $0<μ\leq 1$, the expected maximum is infinite, even for finite values of $n$. For $1<μ\leq 2$, we obtain all the non-vanishing terms in the asymptotic expansion of the expected maximum for large $n$. For $c<0$ and $μ=2$, the expected maximum approaches a non-trivial constant as $n$ gets large, while for $1<μ< 2$, it grows as a power law $\sim n^{2-μ}$. For $c>0$, the asymptotic expansion of the expected maximum is simply related to the one for $c<0$ by adding to the latter the linear drift term $cn$, making the leading term grow linearly for large $n$, as expected. Finally, we derive a scaling form interpolating smoothly between the cases $c=0$ and $c\ne 0$. These results are borne out by numerical simulations in excellent agreement with our analytical predictions.

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arXiv2018

Extreme statistics and index distribution in the classical $1d$ Coulomb gas

We consider a one-dimensional gas of $N$ charged particles confined by an external harmonic potential and interacting via the one-dimensional Coulomb potential. For this system we show that in equilibrium the charges settle, on an average, uniformly and symmetrically on a finite region centred around the origin. We study the statistics of the position of the rightmost particle $x_{\max}$ and show that the limiting distribution describing its typical fluctuations is different from the Tracy-Widom distribution found in the one-dimensional log-gas. We also compute the large deviation functions which characterise the atypical fluctuations of $x_{\max}$ far away from its mean value. In addition, we study the gap between the two rightmost particles as well as the index $N_+$, i.e., the number of particles on the positive semi-axis. We compute the limiting distributions associated to the typical fluctuations of these observables as well as the corresponding large deviation functions. We provide numerical supports to our analytical predictions. Part of these results were announced in a recent Letter, Phys. Rev. Lett. 119, 060601 (2017).

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arXiv2018

High-precision simulation of the height distribution for the KPZ equation

The one-point distribution of the height for the continuum Kardar-Parisi-Zhang (KPZ) equation is determined numerically using the mapping to the directed polymer in a random potential at high temperature. Using an importance sampling approach, the distribution is obtained over a large range of values, down to a probability density as small as 10^{-1000} in the tails. Both short and long times are investigated and compared with recent analytical predictions for the large-deviation forms of the probability of rare fluctuations. At short times the agreement with the analytical expression is spectacular. We observe that the far left and right tails, with exponents 5/2 and 3/2 respectively, are preserved until large time. We present some evidence for the predicted non-trivial crossover in the left tail from the 5/2 tail exponent to the cubic tail of Tracy-Widom, although the details of the full scaling form remains beyond reach.

cond-mat.dis-nn↗

arXiv2018

Periodic Airy process and equilibrium dynamics of edge fermions in a trap

We establish an exact mapping between (i) the equilibrium (imaginary time) dynamics of non-interacting fermions trapped in a harmonic potential at temperature $T=1/β$ and (ii) non-intersecting Ornstein-Uhlenbeck (OU) particles constrained to return to their initial positions after time $β$. Exploiting the determinantal structure of the process we compute the universal correlation functions both in the bulk and at the edge of the trapped Fermi gas. The latter corresponds to the top path of the non-intersecting OU particles, and leads us to introduce and study the time-periodic Airy$_2$ process, ${\cal A}^b_2(u)$, depending on a single parameter, the period $b$. The standard Airy$_2$ process is recovered for $b=+\infty$. We discuss applications of our results to the real time quantum dynamics of trapped fermions.

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arXiv2018

Wigner function of noninteracting trapped fermions

We study analytically the Wigner function $W_N({\bf x},{\bf p})$ of $N$ noninteracting fermions trapped in a smooth confining potential $V({\bf x})$ in $d$ dimensions. At zero temperature, $W_N({\bf x},{\bf p})$ is constant over a finite support in the phase space $({\bf x},{\bf p})$ and vanishes outside. Near the edge of this support, we find a universal scaling behavior of $W_N({\bf x},{\bf p})$ for large $N$. The associated scaling function is independent of the precise shape of the potential as well as the spatial dimension $d$. We further generalize our results to finite temperature $T>0$. We show that there exists a low temperature regime $T \sim e_N/b$ where $e_N$ is an energy scale that depends on $N$ and the confining potential $V({\bf x})$, where the Wigner function at the edge again takes a universal scaling form with a $b$-dependent scaling function. This temperature dependent scaling function is also independent of the potential as well as the dimension $d$. Our results generalize to any $d\geq 1$ and $T \geq 0$ the $d=1$ and $T=0$ results obtained by Bettelheim and Wiegman [Phys. Rev. B ${\bf 84}$, 085102 (2011)].

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arXiv2017

Large deviations

This is a brief pedagogical introduction to the theory of large deviations. It appeared in the ICTS Newsletter 2017 (Volume 3, Issue 2), goo.gl/pZWA6X.

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arXiv2017

Survival Probability of Random Walks and Lévy Flights on a Semi-Infinite Line

We consider a one-dimensional random walk (RW) with a continuous and symmetric jump distribution, $f(η)$, characterized by a Lévy index $μ\in (0,2]$, which includes standard random walks ($μ=2$) and Lévy flights ($0<μ<2$). We study the survival probability, $q(x_0,n)$, representing the probability that the RW stays non-negative up to step $n$, starting initially at $x_0 \geq 0$. Our main focus is on the $x_0$-dependence of $q(x_0,n)$ for large $n$. We show that $q(x_0,n)$ displays two distinct regimes as $x_0$ varies: (i) for $x_0= O(1)$ ("quantum regime"), the discreteness of the jump process significantly alters the standard scaling behavior of $q(x_0,n)$ and (ii) for $x_0 = O(n^{1/μ})$ ("classical regime") the discrete-time nature of the process is irrelevant and one recovers the standard scaling behavior (for $μ=2$ this corresponds to the standard Brownian scaling limit). The purpose of this paper is to study how precisely the crossover in $q(x_0,n)$ occurs between the quantum and the classical regime as one increases $x_0$.

cond-mat.stat-mech↗

arXiv2017

Statistics of fermions in a $d$-dimensional box near a hard wall

We study $N$ noninteracting fermions in a domain bounded by a hard wall potential in $d \geq 1$ dimensions. We show that for large $N$, the correlations at the edge of the Fermi gas (near the wall) at zero temperature are described by a universal kernel, different from the universal edge kernel valid for smooth potentials. We compute this $d$ dimensional hard edge kernel exactly for a spherical domain and argue, using a generalized method of images, that it holds close to any sufficiently smooth boundary. As an application we compute the quantum statistics of the position of the fermion closest to the wall. Our results are then extended in several directions, including non-smooth boundaries such as a wedge, and also to finite temperature.

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arXiv2017

Exact extremal statistics in the classical $1d$ Coulomb gas

We consider a one-dimensional classical Coulomb gas of $N$ like-charges in a harmonic potential -- also known as the one-dimensional one-component plasma (1dOCP). We compute analytically the probability distribution of the position $x_{\max}$ of the rightmost charge in the limit of large $N$. We show that the typical fluctuations of $x_{\max}$ around its mean are described by a non-trivial scaling function, with asymmetric tails. This distribution is different from the Tracy-Widom distribution of $x_{\max}$ for the Dyson's log-gas. We also compute the large deviation functions of $x_{\max}$ explicitly and show that the system exhibits a third-order phase transition, as in the log-gas. Our theoretical predictions are verified numerically.

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