arXiv subjects

Naftali R. Smith

Publications and source records attributed to Naftali R. Smith.

At least 19 recordsLinked to original sources

arXiv2026

The Tracy-Widom distribution at large Dyson index

We study the Tracy-Widom (TW) distribution $f_β(a)$ in the limit of large Dyson index $β\to +\infty$. This distribution describes the fluctuations of the rescaled largest eigenvalue $a_1$ of the Gaussian (alias Hermite) ensemble (G$β$E) of (infinitely) large random matrices. We show that, at large $β$, its probability density function takes the large deviation form $f_β(a) \sim e^{-βΦ(a)}$. While the typical deviation of $a_1$ around its mean is Gaussian of variance $O(1/β)$, this large deviation form describes the probability of rare events with deviation $O(1)$, and governs the behavior of the higher cumulants. We obtain the rate function $Φ(a)$ as a solution of a Painlevé II equation. We derive explicit formula for its large argument behavior, and for the lowest cumulants, up to order 4. We compute $Φ(a)$ numerically for all $a$ and compare with exact numerical computations of the TW distribution at finite $β$. These results are obtained by applying saddle-point approximations to an associated problem of energy levels $E=-a$, for a random quantum Hamiltonian defined by the stochastic Airy operator (SAO). We employ two complementary approaches: (i) we use the optimal fluctuation method to find the most likely realization of the noise in the SAO, conditioned on its ground-state energy being $E$ (ii) we apply the weak-noise theory to the representation of the TW distribution in terms of a Ricatti diffusion process associated to the SAO. We extend our results to the full Airy point process $a_1>a_2>\dots$ which describes all edge eigenvalues of the G$β$E, and correspond to (minus) the higher energy levels of the SAO, obtaining large deviation forms for the marginal distribution of $a_i$, the joint distributions, and the gap distributions.

cond-mat.stat-mech↗

arXiv2025

Remarkable similarities in distributions of dynamical observables in chaotic systems

The study of chaotic systems, where rare events play a pivotal role, is essential for understanding complex dynamics due to their sensitivity to initial conditions. Recently, tools from large deviation theory, typically applied in the context of stochastic processes, have been used in the study of chaotic systems. Here, we study dynamical observables, $A = \sum_{n=1}^N g(\textbf{x}_n)$, defined along a chaotic trajectory $\{\textbf{x}_1, \textbf{x}_2, \ldots, \textbf{x}_N\}$. For most choices of $g(\textbf{x})$, $A$ satisfies a central limit theorem: At large sequence size $N \gg 1$, typical fluctuations of $A$ follow a Gaussian distribution with a variance that scales linearly with $N$. Large deviations of $A$ are usually described by the large deviation principle, that is, $P(A) \sim e^{- N I(A/N)}$, where $I(a)$ is the rate function. We find that certain dynamical observables exhibit a remarkable statistical similarity: even when constructed with distinct functions $g_1(\textbf{x})$ and $g_2(\textbf{x})$, different observables are described by the same rate function. We provide a physical interpretation for this striking similarity by showing that $g_1(\textbf{x})-g_2(\textbf{x})$ belongs to a class of functions that we call ``derived''. Furthermore, we show that if $g(\textbf{x})$ itself is ``derived'', then the distribution of $A$ becomes independent of $N$ in the large-$N$ limit, and is generally non-Gaussian (although it is mirror-symmetric). We demonstrate that the position observable for certain open maps, used to model random walks and the finite-time Lyapunov exponent (FTLE) for the logistic map are of this derived form, thus providing a simple explanation for some existing results.

nlin.CD↗

arXiv2025

Subleading-order theory for condensation transitions in large deviations of sums of independent and identically distributed random variables

We study the full distribution $P_{N}\left(A\right)$ of sums $A = \sum_{i=1}^N$ where $x_1, \dots, x_N$ are $N \gg 1$ independent and identically distributed random variables each sampled from a given distribution $p(x)$ with a subexponential $x \to \infty$ tail. We consider two particular cases: (I) the one-sided stretched exponential distribution $p(x) \propto e^{-x^α}$ where $0 < x < \infty$, (II) the two-sided stretched exponential distribution $p(x) \propto e^{-|x|^α}$ where $-\infty < x < \infty$. We assume $0 < α< 1$ (in both cases). As follows immediately from known theorems, for both cases (i) typical fluctuations of $ΔA = A - \left\langle A\right\rangle $ are described by the central-limit theorem, (ii) the tail $A \to \infty$ is described by the big-jump principle $P_{N}\left(A\right) \simeq N p\left(A\right)$, and (iii) in between these two regimes there is a nontrivial intermediate regime which displays anomalous scaling $P_{N}\left(A\right) \sim e^{-N^βf(ΔA/N^γ)}$ with anomalous exponents $β,γ\in (0,1)$ and large-deviation function $f(y)$ that are all exactly known. In practice, although these theoretical predictions of $P_{N}\left(A\right)$ work very well in regimes (i) and (ii), they often perform quite poorly in the intermediate regime (ii), with errors of several orders of magnitude for $N$ as large as $10^4$. We calculate subleading order corrections to the theoretical predictions in the intermediate regime. We find that for $0 < α< α_c$, these corrections scale as power laws in $N$, while for $α_c < α< 1$ they scale as stretched exponentials, where the threshold value is $α_c = 1/2$ in case (I) and $α_c = 2/3$ in case (II). This difference between the two cases is a result of the mirror symmetry $p(x) = p(-x)$ which holds only in the latter case.

cond-mat.stat-mech↗

arXiv2025

Full distribution of the number of distinct sites visited by a random walker in dimension $d \ge 2$

We study the full distribution $P_M(S)$ of the number of distinct sites $S$ visited by a random walker on a $d$-dimensional lattice after $M$ steps. We focus on the case $d \ge 2$, and we are interested in the long-time limit $M \gg 1$. Our primary interest is the behavior of the right and left tails of $P_M(S)$, corresponding to $S$ larger and smaller than its mean value, respectively. We present theoretical arguments that predict that in the right tail, a standard large-deviation principle (LDP) $P_{M}\left(S\right)\sim e^{-MΦ\left(S/M\right)}$ is satisfied (at $M \gg 1$) for $d\ge2$, while in the left tail, the scaling behavior is $P_{M}\left(S\right)\sim e^{-M^{1-2/d}Ψ\left(S/M\right)}$, corresponding to a LDP with anomalous scaling, for $d>2$. We also obtain bounds for the scaling functions $Φ(a)$ and $Ψ(a)$, and obtain analytical results for $Φ(a)$ in the high-dimensional limit $d \gg 1$, and for $Ψ(a)$ in the limit $a \ll 1$ (describing the far left tail). Our predictions are validated by numerical simulations using importance sampling algorithms.

cond-mat.stat-mech↗

arXiv2025

Nonequilibrium steady state of Brownian motion in an intermittent potential

We calculate the steady state distribution $P_{\text{SSD}}(\boldsymbol{X})$ of the position of a Brownian particle under an intermittent confining potential that switches on and off with a constant rate $γ$. We assume the external potential $U(\boldsymbol{x})$ to be smooth and have a unique global minimum at $\boldsymbol{x} = \boldsymbol{x}_0$, and in dimension $d>1$ we additionally assume that $U(\boldsymbol{x})$ is central. We focus on the rapid-switching limit $γ\to \infty$. Typical fluctuations follow a Boltzmann distribution $P_{\text{SSD}}(\boldsymbol{X}) \sim e^{- U_{\text{eff}}(\boldsymbol{X}) / D}$, with an effective potential $U_{\text{eff}}(\boldsymbol{X}) = U(\boldsymbol{X})/2$, where $D$ is the diffusion coefficient. However, we also calculate the tails of $P_{\text{SSD}}(\boldsymbol{X})$ which behave very differently. In the far tails $|\boldsymbol{X}| \to \infty$, a universal behavior $P_{\text{SSD}}\left(\boldsymbol{X}\right)\sim e^{-\sqrt{γ/D} \, \left|\boldsymbol{X}-\boldsymbol{x}_{0}\right|}$ emerges, that is independent of the trapping potential. The mean first-passage time to reach position $\boldsymbol{X}$ is given, in the leading order, by $\sim 1/P_{\text{SSD}}(\boldsymbol{X})$. This coincides with the Arrhenius law (for the effective potential $U_{\text{eff}}$) for $\boldsymbol{X} \simeq \boldsymbol{x}_0$, but deviates from it elsewhere. We give explicit results for the harmonic potential. Finally, we extend our results to periodic one-dimensional systems. Here we find that in the limit of $γ\to \infty$ and $D \to 0$, the logarithm of $P_{\text{SSD}}(X)$ exhibits a singularity which we interpret as a first-order dynamical phase transition (DPT). This DPT occurs in absence of any external drift. We also calculate the nonzero probability current in the steady state that is a result of the nonequilibrium nature of the system.

cond-mat.stat-mech↗

arXiv2024

Full distribution of the ground-state energy of potentials with weak disorder

We study the full distribution $P(E)$ of the ground-state energy of a single quantum particle in a potential $V(\boldsymbol{x}) = V_0(\boldsymbol{x}) + \sqrtε \, v_1(\boldsymbol{x})$, where $V_0(\boldsymbol{x})$ is a deterministic ``background'' trapping potential and $v_1(\boldsymbol{x})$ is the disorder. We consider arbitrary trapping potentials $V_0(\boldsymbol{x})$ and white-noise disorder $v_1(\boldsymbol{x})$, in arbitrary spatial dimension $d$. In the weak-disorder limit $ε\to 0$, we find that $P(E)$ scales as $P(E) \sim e^{-s(E)/ε}$. The large-deviation function $s(E)$ is obtained by calculating the most likely configuration of $V(\boldsymbol{x})$ conditioned on a given ground-state energy $E$. For infinite systems, we obtain $s(E)$ analytically in the limits $E \to \pm \infty$ and $E \simeq E_0$ where $E_0$ is the ground-state energy in the absence of disorder. We perform explicit calculations for the case of a harmonic trap $V_0(\boldsymbol{x}) \propto x^2$ in dimensions $d\in\left\{ 1,2,3\right\}$. Next, we calculate $s(E)$ exactly for a finite, periodic one-dimensional system with a homogeneous background $V_0(x)=0$. We find that, remarkably, the system exhibits a sudden change of behavior as $E$ crosses a critical value $E_c < 0$: At $E>E_c$, the most likely configuration of $V(x)$ is homogeneous, whereas at $E < E_c$ it is inhomogeneous, thus spontaneously breaking the translational symmetry of the problem. As a result, $s(E)$ is nonanalytic: Its second derivative jumps at $E=E_c$. We interpret this singularity as a second-order dynamical phase transition.

cond-mat.stat-mech↗

arXiv2024

Large deviations in statistics of the local time and occupation time for a run and tumble particle

We investigate the statistics of the local time $\mathcal{T} = \int_0^T δ(x(t)) dt$ that a run and tumble particle (RTP) $x(t)$ in one dimension spends at the origin, with or without an external drift. By relating the local time to the number of times the RTP crosses the origin, we find that the local time distribution $P(\mathcal{T})$ satisfies the large deviation principle $P(\mathcal{T}) \sim \, e^{-T \, I(\mathcal{T} / T)} $ in the large observation time limit $T \to \infty$. Remarkably, we find that in presence of drift the rate function $I(ρ)$ is nonanalytic: We interpret its singularity as dynamical phase transitions of first order. We then extend these results by studying the statistics of the amount of time $\mathcal{R}$ that the RTP spends inside a finite interval (i.e., the occupation time), with qualitatively similar results. In particular, this yields the long-time decay rate of the probability $P(\mathcal{R} = T)$ that the particle does not exit the interval up to time $T$. We find that the conditional endpoint distribution exhibits an interesting change of behavior from unimodal to bimodal as a function of the size of the interval. To study the occupation time statistics, we extend the Donsker-Varadhan large-deviation formalism to the case of RTPs, for general dynamical observables and possibly in the presence of an external potential.

cond-mat.stat-mech↗

arXiv2024

Anomalous scalings of fluctuations of the area swept by a Brownian particle trapped in a $|x|$ potential

We study the fluctuations of the area $A=\int_0^T x(t) dt$ under a one-dimensional Brownian motion $x(t)$ in a trapping potential $\sim |x|$, at long times $T\to\infty$. We find that typical fluctuations of $A$ follow a Gaussian distribution with a variance that grows linearly in time (at large $T$), as do all higher cumulants of the distribution. However, large deviations of $A$ are not described by the ``usual'' scaling (i.e., the large deviations principle), and are instead described by two different anomalous scaling behaviors: Moderately-large deviations of $A$, obey the anomalous scaling $P\left(A;T\right)\sim e^{-T^{1/3}f\left(A/T^{2/3}\right)}$ while very large deviations behave as $P\left(A;T\right)\sim e^{-TΨ\left(A/T^{2}\right)}$. We find the associated rate functions $f$ and $Ψ$ exactly. Each of the two functions contains a singularity, which we interpret as dynamical phase transitions of the first and third order, respectively. We uncover the origin of these striking behaviors by characterizing the most likely scenario(s) for the system to reach a given atypical value of $A$. We extend our analysis by studying the absolute area $B=\int_0^T|x(t)| dt$ and also by generalizing to higher spatial dimension, focusing on the particular case of three dimensions.

cond-mat.stat-mech↗

arXiv2024

Inverse Scattering Method Solves the Problem of Full Statistics of Nonstationary Heat Transfer in the Kipnis-Marchioro-Presutti Model

We determine the full statistics of nonstationary heat transfer in the Kipnis-Marchioro-Presutti lattice gas model at long times by uncovering and exploiting complete integrability of the underlying equations of the macroscopic fluctuation theory. These equations are closely related to the derivative nonlinear Schrödinger equation (DNLS), and we solve them by the Zakharov-Shabat inverse scattering method (ISM) adapted by Kaup and Newell (1978) for the DNLS. We obtain explicit results for the exact large deviation function of the transferred heat for an initially localized heat pulse, where we uncover a nontrivial symmetry relation.

cond-mat.stat-mech↗

arXiv2024

Data-driven analysis of annual rain distributions

Rainfall is an important component of the climate system and its statistical properties are vital for prediction purposes. In this study, we have developed a statistical method for constructing the distribution of annual precipitation. The method is based on the convolution of the measured monthly rainfall distributions and does not depend on any presumed annual rainfall distribution. Using a simple statistical model, we demonstrate that our approach allows for a better prediction of extremely dry or wet years with a recurrence time several times longer than the original time series. The method that has been proposed can be utilized for other climate variables as well.

physics.ao-ph↗

arXiv2024

Exact first-order effect of interactions on the ground-state energy of harmonically-confined fermions

We consider a system of $N$ spinless fermions, interacting with each other via a power-law interaction $ε/r^n$, and trapped in an external harmonic potential $V(r) = r^2/2$, in $d=1,2,3$ dimensions. For any $0 < n < d+2$, we obtain the ground-state energy $E_N$ of the system perturbatively in $ε$, $E_{N}=E_{N}^{\left(0\right)}+εE_{N}^{\left(1\right)}+O\left(ε^{2}\right)$. We calculate $E_{N}^{\left(1\right)}$ exactly, assuming that $N$ is such that the "outer shell" is filled. For the case of $n=1$ (corresponding to a Coulomb interaction for $d=3$), we extract the $N \gg 1$ behavior of $E_{N}^{\left(1\right)}$, focusing on the corrections to the exchange term with respect to the leading-order term that is predicted from the local density approximation applied to the Thomas-Fermi approximate density distribution. The leading correction contains a logarithmic divergence, and is of particular importance in the context of density functional theory. We also study the effect of the interactions on the fermions' spatial density. Finally, we find that our result for $E_{N}^{\left(1\right)}$ significantly simplifies in the case where $n$ is even.

cond-mat.stat-mech↗

arXiv2024

Large deviations in statistics of the convex hull of passive and active particles: A theoretical study

We investigate analytically the distribution tails of the area A and perimeter L of a convex hull for different types of planar random walks. For N noninteracting Brownian motions of duration T we find that the large-L and A tails behave as $\mathcal{P}\left(L\right)\sim e^{-b_{N}L^{2}/DT}$ and $\mathcal{P}\left(A\right)\sim e^{-c_{N}A/DT}$, while the small-$L$ and $A$ tails behave as $\mathcal{P}\left(L\right)\sim e^{-d_{N}DT/L^{2}}$ and $\mathcal{P}\left(A\right)\sim e^{-e_{N}DT/A}$, where $D$ is the diffusion coefficient. We calculated all of the coefficients ($b_N, c_N, d_N, e_N$) exactly. Strikingly, we find that $b_N$ and $c_N$ are independent of N, for $N\geq 3$ and $N \geq 4$, respectively. We find that the large-L (A) tails are dominated by a single, most probable realization that attains the desired L (A). The left tails are dominated by the survival probability of the particles inside a circle of appropriate size. For active particles and at long times, we find that large-L and A tails are given by $\mathcal{P}\left(L\right)\sim e^{-TΨ_{N}^{\text{per}}\left(L/T\right)}$ and $\mathcal{P}\left(A\right)\sim e^{-TΨ_{N}^{\text{area}}\left(\sqrt{A}/T\right)}$ respectively. We calculate the large deviation functions $Ψ_N$ exactly and find that they exhibit multiple singularities. We interpret these as dynamical phase transitions of first order. We extended several of these results to dimensions $d>2$. Our analytic predictions display excellent agreement with existing results that were obtained from extensive numerical simulations.

cond-mat.stat-mech↗

arXiv2024

Optimal finite-differences discretization for the diffusion equation from the perspective of large-deviation theory

When applying the finite-differences method to numerically solve the one-dimensional diffusion equation, one must choose discretization steps $Δx$, $Δt$ in space and time, respectively. By applying large-deviation theory on the discretized dynamics, we analyze the numerical errors due to the discretization, and find that the (relative) errors are especially large in regions of space where the concentration of particles is very small. We find that the choice $Δt = {Δx}^2 / (6D)$, where $D$ is the diffusion coefficient, gives optimal accuracy compared to any other choice (including, in particular, the limit $Δt \to 0$), thus reproducing the known result that may be obtained using truncation error analysis. In addition, we give quantitative estimates for the dynamical lengthscale that describes the size of the spatial region in which the numerical solution is accurate, and study its dependence on the discretization parameters. We then turn to study the advection-diffusion equation, and obtain explicit expressions for the optimal $Δt$ and other parameters of the finite-differences scheme, in terms of $Δx$, $D$ and the advection velocity. We apply these results to study large deviations of the area swept by a diffusing particle in one dimension, trapped by an external potential $\sim |x|$. We extend our analysis to higher dimensions by combining our results from the one dimensional case with the locally one-dimension method.

cond-mat.stat-mech↗

arXiv2024

Confined run and tumble particles with non-Markovian tumbling statistics

Confined active particles constitute simple, yet realistic, examples of systems that converge into a non-equilibrium steady state. We investigate a run-and-tumble particle in one spatial dimension, trapped by an external potential, with a given distribution $g(t)$ of waiting times between tumbling events whose mean value is equal to $τ$. Unless $g(t)$ is an exponential distribution (corresponding to a constant tumbling rate), the process is non-Markovian, which makes the analysis of the model particularly challenging. We use an analytical framework involving effective position-dependent tumbling rates, to develop a numerical method that yields the full steady-state distribution (SSD) of the particle's position. The method is very efficient and requires modest computing resources, including in the large-deviations and/or small-$τ$ regime, where the SSD can be related to the the large-deviation function, $s(x)$, via the scaling relation $P_{\rm st}(x)\sim e^{-s\left(x\right)/τ}$.

cond-mat.stat-mech↗

arXiv2024

Macroscopic fluctuation theory of local time in lattice gases

The local time in an ensemble of particles measures the amount of time the particles spend in the vicinity of a given point in space. Here we study fluctuations of the empirical time average $R= T^{-1}\int_{0}^{T}ρ\left(x=0,t\right)\,dt$ of the density $ρ\left(x=0,t\right)$ at the origin (so that $R$ is the local time spent at the origin, rescaled by $T$) for an initially uniform one-dimensional diffusive lattice gas. We consider both the quenched and annealed initial conditions and employ the Macroscopic Fluctuation Theory (MFT). For a gas of non-interacting random walkers (RWs) the MFT yields exact large-deviation functions of $R$, which are closely related to the ones recently obtained by Burenev \textit{et al.} (2023) using microscopic calculations for non-interacting Brownian particles. Our MFT calculations, however, additionally yield the most likely history of the gas density $ρ(x,t)$ conditioned on a given value of $R$. Furthermore, we calculate the variance of the local time fluctuations for arbitrary particle- or energy-conserving diffusive lattice gases. Better known examples of such systems include the simple symmetric exclusion process, the Kipnis-Marchioro-Presutti model and the symmetric zero-range process. Our results for the non-interacting RWs can be readily extended to a step-like initial condition for the density.

cond-mat.stat-mech↗

arXiv2023

Nonequilibrium steady state of trapped active particles

We consider an overdamped particle with a general physical mechanism that creates noisy active movement (e.g., a run-and-tumble particle or active Brownian particle etc.), that is confined by an external potential. Focusing on the limit in which the correlation time $τ$ of the active noise is small, we find the nonequilibrium steady-state distribution $P_{\text{st}}\left(\mathbf{X}\right)$ of the particle's position $\mathbf{X}$. While typical fluctuations of $\mathbf{X}$ follow a Boltzmann distribution with an effective temperature that is not difficult to find, the tails of $P_{\text{st}}\left(\mathbf{X}\right)$ deviate from a Boltzmann behavior: In the limit $τ\to 0$, they scale as $P_{\text{st}}\left(\mathbf{X}\right)\sim e^{-s\left(\mathbf{X}\right)/τ}$. We calculate the large-deviation function $s\left(\mathbf{X}\right)$ exactly for arbitrary trapping potential and active noise in dimension $d=1$, by relating it to the rate function that describes large deviations of the position of the same active particle in absence of an external potential at long times. We then extend our results to $d>1$ assuming rotational symmetry.

cond-mat.stat-mech↗

arXiv2023

Dynamical phase transition in the occupation fraction statistics for non-crossing Brownian particles

We consider a system of $N$ non-crossing Brownian particles in one dimension. We find the exact rate function that describes the long-time large deviation statistics of their occupation fraction in a finite interval in space. Remarkably, we find that, for any general $N \geq 2$, the system undergoes $N-1$ dynamical phase transitions of second order. The $N-1$ transitions are the boundaries of $N$ phases that correspond to different numbers of particles which are in the vicinity of the interval throughout the dynamics. We achieve this by mapping the problem to that of finding the ground-state energy for $N$ noninteracting spinless fermions in a square-well potential. The phases correspond to different numbers of single-body bound states for the quantum problem. We also study the process conditioned on a given occupation fraction and the large-$N$ limiting behavior.

cond-mat.stat-mech↗

arXiv2023

Striking universalities in stochastic resetting processes

Given a random process $x(τ)$ which undergoes stochastic resetting at a constant rate $r$ to a position drawn from a distribution ${\cal P}(x)$, we consider a sequence of dynamical observables $A_1, \dots, A_n$ associated to the intervals between resetting events. We calculate exactly the probabilities of various events related to this sequence: that the last element is larger than all previous ones, that the sequence is monotonically increasing, etc. Remarkably, we find that these probabilities are ``super-universal'', i.e., that they are independent of the particular process $x(τ)$, the observables $A_k$'s in question and also the resetting distribution ${\cal P}(x)$. For some of the events in question, the universality is valid provided certain mild assumptions on the process and observables hold (e.g., mirror symmetry).

cond-mat.stat-mech↗