Using elementary spectral analysis, a modular random walk algorithm was implemented into an environment with specified absorbing states.

1. RANDOMWALKS,
ABSORBING STATES,
ANDYOU
Kyle Poe | MATH 145 | February 27, 2018

2. APPLICATIONS OF RANDOM WALKS
• A random walk describes a stochastic
process which takes a state into a series
of consecutive following states which
cannot be predetermined.
• Analysis of random walks can give us
insight into important questions like:
• Will a drunk find his way home?
• Will a bee escape a trap?
• How will a gas diffuse?

3. “Given two states from which a
stochastic 2-D system cannot
escape, what is the probability of
ending in either given some starting
configuration?”

4. DESIGN OF A RANDOMWALK PROBABILITY
MATRIX
• For a two-dimensional random walk, we
may consider the grid of states to be
mapped into a single column vector.
• A transformation matrix may then be
designed such that it maps state
probability z into adjacent states with
probability
Where for a non-boundary state, the
probability is uniformly
1
5
.
px,y
px,y+1
px+1,y
px,y-1
px-1,y

5. • We then have the
following general line
from the linear system:
• The following example
depicts the 9 by 9
transition matrix for a
simple 3 by 3 grid:
• Some entries have a
higher value to prevent
diffusing ”off the grid”
y  y+1 X  x+1 (x,y)  (x,y) X  x-1 y  y-1
px,y
px,y+1
px+1,y
px,y-1
px-1,y

6. y+1 x+1 (x,y) x-1 y-1
INTRODUCING ABSORBING STATES
• The definition of an absorbing state is
a state that cannot be left once
arrived at
• The column of the transition matrix
corresponding to the absorbing state
should then simply have a 1 on the
diagonal
• For the earlier 3x3 example, an
absorbing state at (x,y) = (3,2) yields
1
0
0
0
0

7. SPECTRAL DECOMPOSITION
• By design, the matrix has
stable eigenvectors
corresponding to the ”sink”
states
• For holes at (3,2), and (3,3),
we have the following
visualization of the stable
eigenvectors:
• As can be seen in the next
slide, all other eigenvectors
have eigenvalues less than 1,
so all steady states must
therefore be superpositions
of these two states
𝑣1 =
𝑣2 =

8. LONGTERM BEHAVIOR
• We will now direct our attention to a 75 by 75 version of the system with sinks at (10,10) and
(20,20) (much more exciting!)
• Given our knowledge of the behavior of diagonal matrices, the following leads us back to our
earlier conclusion that the final state is a superposition of the sinks:
• This assertion is further supported graphically!

9. LONGTERM BEHAVIOR
-VISUALIZED
• Although the scope of time is too
large to observe in a tolerable
animation, there is indeed
convergence to the distribution
predicted by the method outlined on
the previous slide.
• This kind of analysis can be
performed on ANY random walk
scenario, including those which have
non-equal diffusing possibility