# What is the expected number of steps that a random walker needs to reach a place with high probability in a 2D lattice?

Let $$L_{m,m}$$ be a $$2D$$ lattice. Also, suppose that there is a random walker located in position $$(0,0)$$. The random walker goes right, left, up, or down randomly in each step and cannot get out of $$L_{m,m}$$.

What is the expected number of steps that the random walker needs to reach $$(m,m)$$ with high probability?

• What does the walker do when she hits the boundary? Stay there? Bounce off? – kimchi lover Apr 4 at 11:56
• @kimchilover Bounce off – Hasan Heydari Apr 4 at 13:49
• So edit your question to include this detail. – kimchi lover Apr 4 at 14:17