# Complexity Explorer Santa Few Institute

## Random Walks

• Introduction
• Brownian Motion
• Types of Random Walks
• Root Mean Square Displacement
• Role of the Spatial Dimension
• Part I
• Part II
• Part III
• Part IV
• A poor person's fluctuation dissipation relation
• Part I
• Part II
• Part III
• First Passage Phenomena
• Part I
• Part II
• Final Remarks
• Homework Solutions

#### 6.1 First Passage Phenomena » Quiz Solutions

Question 1

Let  denote the probability for a random walk that starts at site  to reach  without ever reaching

Using the backward Kolmogorov approach, these exit probabilities obey the recursions:

Solving these two equations for the two unknowns gives .

Note that this problem can also be solved by enumerating all paths that take the walk from  to

This enumeration approach becomes impossibly complicated for long intervals, however, while the backward Kolmogorov approach works easily for an interval of any length.

Question 2

Let denote the average time for the random walk to exit the interval  when the walk starts at site $n$.  Again using the backward Kolmogorov approach, these exit times obey the recursions

Solving these two equations for the two unknowns gives .

This problem can also be solved by enumerating all paths that take the walk from to either or .  This enumeration becomes impossibly complicated for long intervals, however, while the backward Kolmogorov approach again works easily for an interval of any length.

Question 3

For the interval the backward Kolmogorov equations for the exit times are:

Solving these equations for the three unknowns gives , and .