Simulations

Simulations

Introduction: Random growth and the Asymmetric Simple Exclusion Process (ASEP)

• The yellow line represents the height function for the random growth profile.

• The blue line represents the initial configuration for the random growth profile.

• The green line represent the limit (expected) shape for the random growth profile.

The rules for the simulation above is described schematically by the picture below:

• Boxes fall on the available corners (i.e. valleys) with rate p.

• Boxes leave on the available corners (i.e. peaks) with rate q.

The height profile may be represented by a sequence of dots on the lattice line and the rules for the simulation may be describe more precisely in that setting:

The particles jump at random times dictated by a random clock with exponential distribution of rate 1.

• The particles have an asymmetric bias when they decide to move so that particles move to the right with probability p and to the left with probability q.

• The particles take simple steps, meaning that particles move by one position at a time.

• The particles obey an exclusion rule so that no two particles occupy the same position at the same time. Hence, the red and green arrows in the diagram above.

• More precisely, this is a Markov process.

This model is called the asymmetric simple exclusion process (ASEP).