We consider planar first-passage percolation and show that the time constant can be bounded by multiples of the first and second tertiles of the weight distribution. As a consequence, we obtain a counter-example to a problem proposed by Alm and Deijfen (2015).

We pursue the study of a random coloring first passage percolation model introduced byFontes and Newman. We prove that the asymptotic shape of this first passage percolationmodel continuously depends on the law of the coloring. The proof uses several couplings,particularly with greedy lattice animals.

The pond turtle (Trachemys scripta elegans) exhibits a notably sluggish pupillary light reflex (PLR), with pupil constriction developing over several minutes following light onset. In the present study, we examined the dynamics of the efferent branch of the reflex in vitro using preparations consisting of either the isolated head or the enucleated eye. Stimulation of the oculomotor nerve (nIII) using 100-Hz current trains resulted in a maximal pupil constriction of 17.4% compared to 27.1% observed in the intact animal in response to light. When current amplitude was systematically increased from 1 to 400 μA, mean response latency decreased from 64 to 45 ms, but this change was not statistically significant. Hill equations fitted to these responses indicated a current threshold of 3.8 μA. Stimulation using single pulses evoked a smaller constriction (3.8%) with response latencies and threshold similar to that obtained using train stimulation. The response evoked by postganglionic stimulation of the ciliary nerve using 100-Hz trains was largely indistinguishable from that of train stimulation of nIII. However, application of single-pulse stimulation postganglionically resulted in smaller pupil constriction at all current levels relative to that of nIII stimulation, suggesting that there is amplification of efferent drive at the ganglion. Time constants for constrictions ranged from 88 to 154 ms with relaxations occurring more slowly at 174–361 ms. These values for timing from in vitro are much faster than the time constant 1.66 min obtained for the light response in the intact animal. The rapid dynamics of pupil constriction observed here suggest that the slow PLR of the turtle observed in vivo is not due to limitations of the efferent pathway. Rather, the sluggish response probably results from photoreceptive mechanisms or central processing.

AC amplifiers can introduce significant distortions into the low frequency and DC components of recorded electrophysiological data such as event-related potentials (ERPs). Methods for correcting such distortions (i.e., estimating the waveform of the original data) after the data have been amplified and recorded rely on an accurate estimate of the amplifier's time constant (TC). We show that the filter characteristics of AC amplifiers in at least some commercially available ERP recording instruments may vary considerably across individual channels, even when each houses an identical AC amplifier circuit. Clearly, distortion correction methods must take this variability into account. We propose an empirical means of estimating the correct TC value. This approach yields more accurate correction than those based on TCs calculated analytically.

Studies of concurrent central, and autonomic activity using a conventional event-related potential (ERP) oddball paradigms, are considered useful in elucidating the relationship between central and autonomic responses, but the autonomic response tends to overlap. A new method was used to decompose and score overlapping skin conductance responses (SCR). This method enabled examination of dynamic relationships of phasic SCR, prestimulus electroencephalogram (EEG), and ERP to auditory target stimuli in 50 normal adults. SCR amplitude was negatively correlated to EEG and N200 amplitude. The SCR amplitude changes over time exhibited an exponential decline opposite to those of N200, alpha, and beta. All the fitted exponential functions had a time constant of 1–2 min. The findings suggest that a N200 component, active in the auditory sensory discrimination, is concomitant with the SCR. The narrow range of the time constant may provide a clue to the conjoint processes underlying central and autonomic adaptive functions.

We consider first-passage percolation in an infinite horizontal strip of finite height. Using methods from the theory of Markov chains, we prove a central limit theorem for first-passage times, and compute the time constants for some special cases.

Let U be the distribution function of the non-negative passage time of an individual edge of the square lattice, and let a0n be the minimal passage time from (0, 0) to (n, 0). The process a0n/n converges in probability as n → ∞to a finite constant μ (U) called the time constant. It is proven that μ (Uk)→ μ(U) whenever Uk converges weakly to U.

Let U be the distribution function of the passage time of an individual bond of the square lattice, and let pT be the critical probability above which the expected size of the open component of the origin (in the usual bond percolation) is infinite. It is shown that if (∗)U(0–) = 0, U(0) < pT, then there exist constants 0 < a, C1 < ∞ such that a self-avoiding path of at least n steps starting at the origin and with passage time ≦ an} ≦ 2 exp (–C1n).

From this it follows that under (∗) the time constant μ (U) of first-passage percolation is strictly positive and that for each c > 0 lim sup (1/n)Nn(c) <∞, where Nn(c) is the maximal number of steps in the paths starting at the origin with passage time at most cn.

Let μ (F) be the time constant of first-passage percolation on the square lattice with underlying distribution function F. Two theorems are presented which show, under some restrictions, that μ varies continuously in F with respect to weak convergence. These results are improvements of existing continuity theorems.

A generalization of first-passage percolation theory proves that the fundamental convergence theorems hold provided only that the time coordinate distribution has a finite moment of a positive order. The existence of a time constant is proved by considering first-passage times between intervals of sites, rather than the usual point-to-point and point-to-line first-passage times. The basic limit theorems for the related stochastic processes follow easily by previous techniques. The time constant is evaluated as 0 when the atom at 0 of the time-coordinate distribution exceeds½.