Perron's theorem has many applications. I will explain three. As Perron's theorem applies to any positive square matrix, we can use the theorem in any application pertaining to positive square matrices. Probabilities and ratios are common elements in these matrices as they are strictly nonnegative.

Markov Chains

Markov Chain: A system that transitions from one state to another based solely on the current state ("Markov Chain", Anton).

Eugene Onegin Example

Markov’s theorem can be employed for almost any repeated experiment where one outcome depends on the previous outcome.

The final state vector is in fact the eigenvector associated with the Perron root for these special cases. Hence, Perron's Theorem is extremely helpful for proving Markov's Theorem.

Leslie Model

Rabbit Problem

This is a simple population analysis based on rabbit birth and death rates.

The recursion is a follows:

  • We start with one pair of youth rabbits

  • After one month, the original pair become adults and gives birth to a pair

  • The next month, the original pair dies and the youth pair ages up and give birth to a pair

  • Repeat

Leslie population models are nonnegative and irreducible so it turns out that their steady state (i.e. f(t) as t approaches infinity) is the Perron root by Markov's theorem.

Page Rank

Sergey Brin and Larry Page developed Google’s PageRank. It is very well know now, even acknowledging that it was not the first page ranking algorithm. The algorithm is designed to place value on a webpage based on the links between pages (Emerging Technology from the arXiv). This algorithm gets very complicated, quickly. Perron-Frobenius theory lets us eliminate some time consuming matrix power calculations.

Ranking Example

The resulting steady state assigns page 1 the highest rank even though page 3 is linked to more. As we intended, the page rank was affected by weight of a page's link (Tanase).

In fact, more than 4 pages exist. So, through a little manipulation, the developers of the page rank algorithm arrived at an A' which is a sparse, positive, primitive, convergent Markov chain.

Now, we can solve the problem using eigenvectors guaranteed by the Perron-Frobenius theorem instead of a lot of matrix multiplication. However this may not be very fast either. That is why Brin and Page used the previously mentioned iteration until convergence was reached. This convergence is guaranteed by Markov’s theorem. In fact, a good approximation is reached fairly quickly (Tanase).

Other Applications

Similar to the Ehrenfest Experiment, the diffusion lab shows a model where molecules start out in separate boxes with associated probabilities of transferring boxes due to the laws of physics. The closed system eventually reaches a steady state that can be calculated using Perron's theorem and Markov's theorem.

To start, try the simulation with the same number of molecules in each side and increase the temperature in one box. you can use the stopwatch to time the mixing after removing the divider. Use the pause button to check the molecule distribution at any time.

In a one-dimensional random walk, we move one of two directions at each step, randomly. This creates a Markov chain where our dot has a placement depending on the previous step with a positive probability determining the next step. For this simulation, set the particles to an even number and be sure to use the fixed step configuration. Then, click the Walk 1D button. At each step, the particle has equal probability of moving either up or down. Hence, we expect the average amount of steps up and steps down to be equal.