# Random walk with an absorbing final state

Consider a virtual beacker constitued of $$500\times 500$$ loci, all initally at the stage $$0$$. Each of these loci is subject to a discrete in time random walk through $$n$$ ordered other stages, with forward and backward evolutions, and with an absorbing last stage.

$\begin{matrix} 0 & \rightleftharpoons[d]{u} & 1 & \rightleftharpoons[d]{u} & 2 & \dots & \rightarrow[]{u} & n\\ \end{matrix}$ More precisely, being given a locus at the stage $$i$$, ($$1\leq i\leq n-1$$),
• with probability $$u$$, it evolves forward to the stage $$i+1$$,
• with probability $$d$$, it evolves backward to the stage $$i-1$$,
• with probability $$1-(u+d)$$, it remains at stage $$i$$.
Loci at the stage $$0$$ evolve
• with probability $$u$$ to the stage $$1$$,
• with probability $$1-u$$, it remains at stage $$0$$.
Loci at the final absorbing stage $$n$$ do not evolve anymore and are figured with spots.

Try yourself !
1. Choose a set of parameters (with $$u+d<1$$).
2. Click 'Complex formation' to initiate the evolution of the beacker.
3. Click 'Degree of occupancy' to count the amount of loci in each stages.
4. Click 'Number of rejections' to follow the evolution of the average number of times each loci reaches the initial stage 0.

The case of a competitive binding with two chain of evolution is available here.
Simulation based on Simulab.