There are 1008 tiles arranged as 84 rows and 12 columns. Each tile is an independently sampled worry frog from 93 unique worry frogs. Adjacency is understood to be cardinal adjacency, which excludes diagonal adjacency.

• For any worry frog not on the boundary, what is the probability that at least one of its neighbors is the same worry frog?
• What is the probability that any two or more similar worry frogs are on the same row?
• What is the probability that a row has twelve unique worry frogs?
• Clicking on a tile swaps the worry frog to a randomly sampled worry frog that is not itself. A connected component is any cluster of the same worry frog that are adjacent to each other. What is the optimal strategy to connect the entire graph (that is, no isolated vertices) in the least number of clicks? Following this strategy, what is the expected number of clicks?
• You start off with one worry frog that will reproduce asexually over generations. Let \(Z\) denote the number of children for the next generation. Write the probability mass function for \(Z\) as \[ p_i = \mathbb{P}(Z = i),\quad i \geq 0. \] Each child will in turn reproduce independently according to the same distribution, so on and so forth. Prove that whether the population of worry frogs explodes or becomes extinct is entirely dependent on \(m = \mathbb{E}[Z]\) and not on the pmf of \(Z\).
• Let there be \(n\) worry frogs with \(p\) the probability of there being an edge between two worry frogs, (the adjacency in this problem is independent of two frogs being the same or not). Thus, the expected degree of our worry frog graph is \(\delta = (n-1)p\). Show that if \(\delta = \kappa \log n\) for some \(\kappa > 1\), we have that for any two worry frogs, there exists at least one path connecting them in probability as \(n \xrightarrow{} \infty\). Furthermore, show that for \(\kappa \in (0,1)\), there exists at least one pair of worry frogs for which there are no paths connecting them in probability, as \(n \xrightarrow{} \infty\).
• For \(pn^2 = o(\sqrt{n})\), show that with probability tending to \(1\), no connected component has three or more worry frogs. Furthermore, let \(\delta = o(1)\). Show that as \(n \xrightarrow{} \infty\), the probability that there is a cycle in the worry graph is zero.
• Let infinitely many worry frogs be scattered as vertices of equilateral triangles arranged in a honeycomb tiling in 2D. For every pair of worry frogs, let \(p\) denote the probability of there being and edge between them if they are in the same triangle, that is, edges can only be formed between worry frogs that are 1 hop apart. Show that there exists a \(p_1 > 0.2\) such that for \(p < p_1\), the probability of there being an infinite connected component is zero. Show that there exists a \(p_2 < 1\) such that for \(p > p_2\), the probability of there being an infinite connected component is one.
• HARD: Consider the \(Z^2\) lattice where the probability of there being a worry frog at any point in the lattice is \(q\). Two worry frogs are connected if their Euclidean distance is 1. Find \(q_c\) such that for \(q \leq q_c\), the probability of there being an infinite connected component is zero, while for \(q > q_c\), the probability of there being an infinite connected component is one.