Extreme event statistics in a drifting Markov chain

We analyze extreme event statistics of experimentally realized Markov chains with various drifts. Our Markov chains are individual trajectories of a single atom diffusing in a one-dimensional periodic potential. Based on more than 500 individual atomic traces we verify the applicability of the Sparre Andersen theorem to our system despite the presence of a drift. We present detailed analysis of four different rare-event statistics for our system: the distributions of extreme values, of record values, of extreme value occurrence in the chain, and of the number of records in the chain. We observe that, for our data, the shape of the extreme event distributions is dominated by the underlying exponential distance distribution extracted from the atomic traces. Furthermore, we find that even small drifts influence the statistics of extreme events and record values, which is supported by numerical simulations, and we identify cases in which the drift can be determined without information about the underlying random variable distributions. Our results facilitate the use of extreme event statistics as a signal for small drifts in correlated trajectories.

Figure 1

Overview of the experiment and its related quantities. (a) Schematic view of the experimental setup. A Cs atom is trapped in a one-dimensional blue-detuned optical lattice, which is overlapped with an optical dipole trap for radial confinement. The diffusion of the atom is initiated by six beams forming a three-dimensional optical molasses. This light field is also used for fluorescence imaging of the atoms to infer their position in the optical lattice. (b) A typical fluorescence trace of the diffusing atom consisting of 14 individual images. The corresponding position trace is depicted in panel (c). Here, squares (circles) depict lower (upper) records of the trace and the filled symbols indicate the extreme values obtained. Out of hundreds of such traces, we can calculate on one hand the distance distribution $\varphi \left(\xi \right)$ for different flight times $t_{\mathrm{flight}}$, shown exemplarily for (d) $t_{\mathrm{flight}}=1\mathrm{ms}$ and (e) $t_{\mathrm{flight}}=50\mathrm{ms}$. While for small $t_{\mathrm{flight}}$ almost no shift of the mean value (solid vertical lines) from the origin (dashed lines) is detected in the distance distribution, it becomes visible for larger $t_{\mathrm{flight}}$. (f) Mean values of the position distribution $\langle \psi \left(x_k\right)\rangle$ for increasing step $k$ for $t_{\mathrm{flight}}=0.1\mathrm{ms}$ (blue $\circ$), $t_{\mathrm{flight})}=1\mathrm{ms}$ (green $▵$), $t_{\mathrm{flight})}=10\mathrm{ms}$ (purple $\square$), and $t_{\mathrm{flight})}=50\mathrm{ms}$ (orange $\diamond$). (g) Position distributions $\psi \left(x_k\right)$ for $t_{\mathrm{flight}}=0.1\mathrm{ms}$ with $k=2$ and (h) $k=13$, illustrating the increasing importance of the drift with increasing step number $k$. The dashed lines in panels (g) and (h) again indicate the origin, while solid vertical lines depict the mean values of the position distributions.

Figure 2

First passage time probability at step $k$ for an atom starting its random walk at the origin $x_0=0$. The inset depicts the first passage from the negative to the positive semi-axis of the atom of which the full trajectory is shown in Fig. 1 c. Data points are for $t_{\mathrm{flight}}=1\mathrm{ms}$ (green triangles) and $t_{\mathrm{flight}}=50\mathrm{ms}$ (orange diamonds). The probability is given by the Sparre–Andersen formula see Eq. (2) depicted as the black solid line, which fits equally well to both data sets. The dashed gray line depicts the large-$k$ limit. Error bars are standard statistical errors.

Figure 3

Measured (blue, filled) distributions of (a) $M_{\min }$, (b) $M_{\max }$, (c) $R_{\min }$, and (d) $R_{\max }$. The solid (orange) line indicates the result of numerical simulations in good agreement with the measured data and the dashed (red) line is an exponential fit to the data. Because the atom is more likely to cross the origin directly in the beginning for a system with drift, as reflected by the first passage probability (see Fig. 2), $M_{\max }$ experiences a large peak at zero. Therefore the fit does not perfectly resemble the distribution. However, when excluding the zero peak, the data are still described by an exponential distribution; see dash-dotted (green) line.

Figure 4

Position distribution $\psi \left(x_k\right)$ (gray, filled) plotted together with the distribution of $M_{\min }$ (dark blue solid line) and $R_{\min }$ (yellow solid line) in (a) and $M_{\max }$ (dark blue solid line) and $R_{\max }$ (yellow solid line) in (b), respectively. The same is shown in (c,d), but for a trace length of $n=14$. While the position distribution $\varphi \left(x_k\right)$ contains all positions an atom occupied in step $k$, the extreme value distribution includes only the maximum (in positive or negative direction) reached by the atom until step $k$. The record value distribution is further connected with the extreme value distribution as the last record (value) achieved is always the extreme value. Moreover, the first value reached on the negative (positive) side counts as the first lower (upper) record value, as it is always smaller (greater) than zero.

Figure 5

(a) Comparison of $\langle M_{\min ,\max }\rangle$ (blue $\square$), $\langle R_{\min ,\max }\rangle$ (orange $\diamond$), and $\langle \psi \left(x_k\right)\rangle$ (light gray $\circ$) for $t_{\mathrm{flight}}=1\mathrm{ms}$. (b) Differences of the mean of the extreme values distributions ${\delta }_{\mathrm{M}}$ and of the mean of the record value distributions ${\delta }_{\mathrm{R}}$ for increasing trace length. The solid (blue) line is a linear fit to ${\delta }_{\mathrm{M}}$ overlapping with a linear fit to $\langle \psi \left(x_k\right)\rangle$ (dashed gray line).

Figure 6

(a) Simulated expectation value of $\langle M_{\min }\rangle /\sigma$ (color bar) and of (b) $\langle R_{\min }\rangle /\sigma$ (color bar) for increasing ${\mu }_{\mathrm{rel}}$ and trace length $n$. Dashed vertical lines are cuts at specific ${\mu }_{\mathrm{rel}}$ depicted in panels (c) and (d) and horizontal dashed lines are cuts at specific trace lengths $n$ depicted in panels (e) and (f). The data points are measured values for ${\mu }_{\mathrm{rel}}=0.1$ corresponding to $t_{\mathrm{flight}}=1\mathrm{ms}$ and the dashed blue line in panel (c) indicates the theoretical prediction for a system without drift; cf. Eq. (3). A linear increase is found for $\langle M_{\min }\rangle /\sigma$ (and $\langle R_{\min }\rangle /\sigma$, respectively) with increasing ${\mu }_{\mathrm{rel}}$, which is indicated by the dotted lines in panels (e) and (f).

Figure 7

Measured occurrences of maxima (dark blue) and minima (light gray) for (a) $t_{\mathrm{flight}}=0.1\mathrm{ms}$, (b) 1 ms, (c) 10 ms, and (d) 50 ms for fixed $n=14$. Solid lines indicate the results from numerical simulations which are in good agreement with the data measured. The dashed red line indicates the result derived from the Sparre–Andersen theorem in Eq. (4).

Figure 8

(a) Contrast extracted from occurrences of maxima and minima for $t_{\mathrm{flight}}=0.1\mathrm{ms}$ (blue $\circ$), and 50 ms (orange $\diamond$), and $n=14$ [as shown in Figs. 7 and 7]. Solid lines indicate the results from numerical simulations. (b) Data points depict the maximal contrast extracted from the data in panel (a), while symbols and colors match those in Fig. 1. The solid line again indicates the result from numerical simulations and the dashed line depicts a linear fit to the simulation.

Figure 9

Record number distributions shown for various trace length $n$ stated left. Experimental data for upper (blue, left) and lower (light gray, right) records is depicted. The solid lines are numerical simulations and the dashed line represents the analytic result for a system without drift [Eq. (5)]. The dotted lines in the left panel indicate a Gaussian fit to the data according to Eq. (7) of which the constants $a\left(\mu \right)$ and $b\left(\mu \right)$ are extracted. For details see text.

Figure 10

(a) Measured mean record number and (b) variance of the record number for lower (light gray $\circ$) and upper (blue $▵$) records depicted for increasing trace length $n$. The dashed line indicates the theoretical expectation for systems without drift [cf. Eq. (6)]. The solid line depicts the expected values for $\langle {\rho }_{\min }\rangle$ and $\langle {\rho }_{\min }^2\rangle -{\langle {\rho }_{\min }\rangle }^2$, respectively, and the dotted line depicts the theoretical saturation value for $\langle {\rho }_{\max }\rangle$. All three quantities are described by the constants $a\left(\mu \right)$ and $b\left(\mu \right)$, which are extracted from the data of Fig. 9. For details see text.