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3 edited body

Here's a simple birth model which leads to power law behaviour with exponent 1.

Reproduction as follows:

(a) each individual produces "clone offspring" (a child of the same type as itself) at rate 1.

(b) in addition, each individual of type 1 produces "mutant offspring" (a child of a new type not yet seen before) at rate $\mu$, where $\mu$ is any positive constant. So the first mutant will be called type 2, the second type 3, etc.

Let $N_k(t)$ be the number of individuals of type $k$ alive at time $t$.

Once the first individual of type $k$ has been born, the type-$k$ family grows exponentially. Also, the first individual of type $k$ is born at time $\log k + O(1)$.

From this it's quite easy to obtain that $N_k(t)$ behaves something like $e^t/k$. More precisely, for any $k$ the quantity $ke^{-t}N_k(t)$ converges as $t\to\infty$ with probability 1 to some random variable $W_k$, say, and the sequence of distributions of $W_k, k\geq 1$ is tight.

Reordering the $N_k(t)$ into decreasing order still leaves essentially the same rate of decay.

So for large $t$, $N_k(t), k \geq 1$ obeys Zipf's law (for a range of $k$ that depends suitably on $t$, say $k < < \ll e^t$ ).

2 edited body

Here's a simple birth model which leads to power law behaviour with exponent 1.

(b) in addition, each individual of type 1 produces "mutant offspring" (a child of a new type not yet seen before) at rate $\mu$, where $\mu$ is any positive constant. So the first mutant will be called type 2, the second type 3, etc.
Let $N_k(t)$ be the number of individuals of type $k$ alive at time $t$.
Once the first individual of type $k$ has been born, the type-$k$ family grows exponentially. Also, the first individual of type $k$ is born at time $\log k + O(1)$.
From this it's quite easy to obtain that $N_k(t)$ behaves something like $e^t/k$. More precisely, for any $k$ the quantity $ke^{-t}N_k(t)$ converges as $n\to\infty$ t\to\infty$with probability 1 to some random variable$W_k$, say, and the sequence of distributions of$W_k, k\geq 1$is tight. Reordering the$N_k(t)$into decreasing order still leaves essentially the same rate of decay. So for large$t$,$N_k(t), k \geq 1$obeys Zipf's law (for a range of$k$that depends suitably on$t$, say$ k < < e^t $). 1 Here's a simple birth model which leads to power law behaviour with exponent 1. Start with a single individual of type 1. Reproduction as follows: (a) each individual produces "clone offspring" (a child of the same type as itself) at rate 1. (b) in addition, each individual of type 1 produces "mutant offspring" (a child of a new type not yet seen before) at rate$\mu$, where$\mu$is any positive constant. So the first mutant will be called type 2, the second type 3, etc. Let$N_k(t)$be the number of individuals of type$k$alive at time$t$. Once the first individual of type$k$has been born, the type-$k$family grows exponentially. Also, the first individual of type$k$is born at time$\log k + O(1)$. From this it's quite easy to obtain that$N_k(t)$behaves something like$e^t/k$. More precisely, for any$k$the quantity$ke^{-t}N_k(t)$converges as$n\to\infty$with probability 1 to some random variable$W_k$, say, and the sequence of distributions of$W_k, k\geq 1$is tight. Reordering the$N_k(t)$into decreasing order still leaves essentially the same rate of decay. So for large$t$,$N_k(t), k \geq 1$obeys Zipf's law (for a range of$k$that depends suitably on$t$, say$ k < < e^t \$ ).