@@ -24,6 +24,8 @@ to describe a polymer by an ordered set of points $\{\mathbf{r}_0, \mathbf{r}_1,

To simplify the model even further, we will assume that the angle between different bonds is restricted to be a multiple of $90^{\circ}$. In doing so, we restrict the polymer points $\mathbf{r}_i$ to be on a square lattice with lattice constant $a$. Without restricting generality, we can set $a=1$.

![self-avoiding random walk](figures/polymer-selfavoiding.svg)

In the above example, you may already see the last ingredient for our toy model: Since a polymer has a finite extent in space, two subunits cannot come too close. In our lattice model we implement this by demanding that one lattice point cannot be occupied by two different subunits. In other words: $\mathbf{r}_i \neq \mathbf{r}_j$ for all $i \neq j$.

With these assumptions, our polymer model becomes a *self-avoiding random walk on a square lattice*.

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@@ -38,6 +40,8 @@ where $\nu = 3/4$ for 2D, and $\nu\approx 3/5$ in 3D.

The behavior of the self-avoiding random walk should be contrasted to the free random walk where both backtracking (the random walk takes a step back: $\mathbf{r}_{i+2} = \mathbf{r}_i$) and intersections are allowed:

![free random walk](figures/polymer-freerandomwalk.svg)

For the free random walk the scaling behavior is well-known and we have $\nu = 1/2$.

## Why sampling polymers is hard

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@@ -62,6 +66,8 @@ You don't have to do this for the final report ;-)

The Rosenbluth method generates $N$ different polymers/self-avoiding random walks independently from each other, by growing every polymer individually. The polymer is grown successively by adding a new subunit $\mathbf{r}_i$to the end of the polymer, choosing randomly one of the $m_i$ unoccupied lattice sites adjacent to $\mathbf{r}_{i-1}$. You can then encounter different situations:

![adding a new subunit to a polymer](figures/polymer-build.svg)

Note that in the latter case $m_i=0$ no new subunit can be added, and the polymer sampling cannot continue beyond a certain length.

This process is repeated $N$ times. For each polymer $k$ (with $k$ denoting the polymer index, $k=1,\dots, N$) we record all positions $\{\mathbf{r}_{k, 1}, \dots, \mathbf{r}_{k, L}\}$ (we can let all polymers start from the same initial point $\mathbf{r}_{k, 0}$). In addition, we record for every polymer the *weight*