transition_prob

Sampling from Transition Probability

Description

Let consider following one step transition probability :

$$ W(il-jl,\,\epsilon) = \begin{cases} \frac{1}{2} & (|i-j| = 1) \\ 0 & \text{else} \end{cases} $$

With assumption of Markov chain, we can get $n$ step transition probability :

$$ W(il-jl,\,n\epsilon) = \begin{cases} 0 & (|i-j| > n \text{ or } i-j+n \text{ is odd}) \\ \frac{1}{2^n}\begin{pmatrix} n \\ \frac{i-j+n}{2} \end{pmatrix} & (|i-j| \leq n \text{ and } i+j-n \text{ is even}) \end{cases} $$

Then in this project, we compare distribution of direct sampling from this transition probability and distribution of end points of random walks.

Setting

• Initial point : $j=0$
• Length of path : $n=100$
• Total number of trials : $N = 100000$

Build Process

# Data Generation
cargo run --release

# Plot
python nc_hist.py

Result

References

• M. Chaichian, A. Demichev, Path Integrals in Physics: Volume I Stochastic Processes and Quantum Mechanics, CRC Press (2001)