Survival-time statistics for sample space reducing stochastic processes

Stochastic processes wherein the size of the state space is changing as a function of time offer models for the emergence of scale-invariant features observed in complex systems. I consider such a sample-space reducing (SSR) stochastic process that results in a random sequence of strictly decreasing integers $\left\{x\right(t\left)\right\},0\le t\le \tau$, with boundary conditions $x\left(0\right)=N$ and $x\left(\tau \right)$ = 1. This model is shown to be exactly solvable: ${\mathcal{P}}_N\left(\tau \right)$, the probability that the process survives for time $\tau$ is analytically evaluated. In the limit of large $N$, the asymptotic form of this probability distribution is Gaussian, with mean and variance both varying logarithmically with system size: $\langle \tau \rangle \sim lnN$ and ${\sigma }_{\tau }^2\sim lnN$. Correspondence can be made between survival-time statistics in the SSR process and record statistics of independent and identically distributed random variables.

Figure 1

Stochastic SSR process as a directed random hopping with variable step size.

Figure 2

Pictorial representation of all possible paths that originate with system size $N=4$ for the SSR process. A dice with $N=4$ faces is thrown at $t=0$; the possible outcomes are 1, 2, 3, and 4, each occurring with equal probability 1/4. A typical path might be, for example, $4\to 3\to 2\to 1$ with survival time $\tau =3$, i.e., the outcome at the first time step is $x(t=1)=3$, then a dice with 2 faces will be selected and it is thrown. Now, the possible outcome would be either 1 or 2 with equal probability 1/2. If the outcome is $x(t=2)=2$, then finally at next time step the outcome must be $x(t=3)=1$ with unit probability.

Figure 3

Survival probability distribution. Shaded circle denotes data obtained from simulation with $N=10$ and total number of samples ${10}^8$, and open triangle corresponds to theoretical curve computed using Eq. (8).

Figure 4

Data collapse curves for the survival probability distribution with different system size $N$ calculated using Eq. (8). Here $v=(\tau -\langle \tau \rangle )/{\sigma }_{\tau }$ and $J\left(v\right)=\sqrt{2\pi {\sigma }_{\tau }^2}{\mathcal{P}}_N\left(\tau \right)$. The continuous curve represents the function $J\left(\nu \right)$ calculated using normalized probability distribution [see Eq. (19)] with $N={10}^8$.

Figure 5

The plots show the mean and variance of survival time, i.e., $\langle \tau \rangle$ and ${\sigma }_{\tau }^2$ as a function of system size $N$. Each point is obtained averaging over ${10}^8$ independent realizations of the SSR process. $N$ is chosen in steps $2^i$ where $i$ runs from 2 to 20. The difference of mean and variance is $\langle \tau \rangle -{\sigma }_{\tau }^2\approx 1.647$ for large $N=2^{20}$, and this number is approximately equal to $\zeta \left(2\right)={\pi }^2/6\approx 1.646$, where $\zeta \left(n\right)$ is the Riemann Zeta function. The straight lines, corresponding to the mean and variance, are computed using theoretical form [see Eq. (18)].