ANIRUDDHAN”S TALKKKMKKKKK

• Percolation Theory

  • Consider the lattice ${\mathbb{Z}}^2$ that gives you a square grid and each edge is open w probability is $p$
  • Let $\theta (p)$ is the probability that the connected component that contains the origin is infinite.
    • Intuitively this should be an increasing function.
  • $p_c:=\text{inf}\{p>0|{\theta }_p>0\}=\frac{1}{2}$
  • Idea is that we uniformaly assign a value to each edge, and close edge if value is more that $p$. Coupling argument.
  • Extension to ${\mathbb{Z}}^d$, all corresponding $p_c^{(d)}$ are strictly
  • theta grows lineary from pc and drops exponentially as it goes below it. so there is some notion of a sharp phase transition for a lattice of a finite size.
  • What if origin is not any special vertex, then what is the probability of there existing an infinite cluster in the lattice.
    • If $p>p_c$ then prob of inf cluster is $1$.
    • If $p<p_c$ then prob of inf cluster if $0$.
  • Now we look at the dual of a graph.