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The fundamental result that completely characterizes recurrent/transient graphs is that a graph is recurrent if and only if the effective resistance of the graph, when considered as an electric networ …

This model (at least when all the $\lambda_{ij}$ are the same and all the $\mu_i$ are the same) is called the Contact Process.

This is an interesting variation of First passage percolation (see this question). There's no reason to think that in this case the answer will be nice in any way, as percolation thresholds rarely are …

For any vertex $v$, you can take $\mu$ to be $7/8 \pi + 1/8 v$, that is, the with probability $1/8$ pick $v$ and otherwise pick a random vertex according to $\pi$. Now the distance to stationarity at …

Consider for a moment the case $m=cn$ for some constant $c$. This also yields a maximal load of $(1+o(1))\log(n)/\log(\log(n))$. Geometric distribution with mean $n$ will give you $cn$ balls with high …

What is your question? In the situation you describe, both distributions are approximately Poisson with the same parameter, so it is no wonder they also approximate each other.

Partition the unit square into small squares of area roughly $a$. Your question is equivalent to asking for which $a$ do we typically see about 1 small square with points from each of $X$,$Y$ and $Z$? …

There are several other models proved to converge to SLE: critical percolation on the triangular lattice, Gaussian Free Field, Harmonic Explorer, and recently also the critical Ising model. You can ch …

The answer is yes. To see this, take a transient graph $G$ (say $\mathbb{Z}^3$) and glue a copy of $\mathbb{N}$ to some vertex (say $v_0$). The vertices of $\mathbb{N}$ now all have $g(v)=1$. If you …

Robert's answer is correct, if you want an asymptotic answer. However, for the question as stated in the edit (that is for all $n\ge k$...) the answer is no. Take for example the case of $p=\frac12$. …

$\newcommand{\E}{\mathbb{E}}$ You practically answered your own question. Denote by $A_k$ the event "$X_0 > k$". Then $X_0=\sum_{k=0}^{m-1} \delta(A_k)$. Denote by $B_k$ the event "$X_k > k$". Then …

When tossing $n$ balls uniformly and independently into $n$ bins, the distribution of the number of balls in any specific bin is binomial with parameters $n$ and $p=\frac{1}{n}$. Thus, the probability …

This is the Balls and Bins setting.

The crucial requirement is that $\sum_{i=0}^\infty t_i^2 < \infty$. See Kolmogorov's two-series theorem and also the more general Kolmogorov's three-series theorem.

In the setting you describe, there's generally no CLT. Let me describe a counterexample showing that $\sum_{i=1}^{k-1} X_i / \sqrt{\sum_{i=1}^{k-1} \sigma_i}$ doesn't tend to normal. I'm pretty sure t …