minimal code for part 1.

Link.py

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+class Link:
+    def __init__(self, direction):
+        """
+        An incompressible link of fixed length a and either a positive or
+        negative direction.
+
+        Parameters:
+            direction (int) The direction. Must be 1 or -1.
+        """
+
+        self.direction = direction

RubberBand.py

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+import numpy as np
+
+from Link import Link
+
+class RubberBand:
+    def __init__(self, N, a=1.):
+        """
+        RubberBand is a class that can simulate a rubber band with N
+        incompressible links of length a.
+
+        Parameters:
+            N (int)     The number of links.
+            a (float)   The fixed length of links.
+        """
+
+        self.N = N
+        self.a = a
+        self.links = self.__sample_links()
+
+    @property
+    def length(self):
+        return self.a * np.sum([l.direction for l in self.links])
+
+    def __sample_links(self):
+        """
+        Sample N Link objects with a random direction.
+        """
+
+        samples = np.random.random_sample((self.N,))
+        directions = np.ones(samples.shape)
+        directions[samples < 0.5] = -1.
+        directions = directions.astype(int)
+        return [Link(d) for d in directions]
+
+

main.py

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+import numpy as np
+import scipy as scp
+
+from matplotlib import pyplot as plt
+
+from RubberBand import RubberBand
+
+NUM_BANDS = int(1E5)
+N = 100
+
+length_dist = np.zeros((NUM_BANDS,))
+for i in range(NUM_BANDS):
+    band = RubberBand(N, a=1)
+    length_dist[i] = band.length/band.a
+
+P_true = lambda N, L: scp.special.binom(N, (L+N)/2) / 2**N
+L_norm = np.linspace(-N, N, 100) # normalised (no Link length a) range
+
+plt.plot(L_norm, P_true(N, L_norm))
+plt.hist(length_dist, range=(-N, N), bins=N, density=True)
+plt.xlabel("$L/a$")
+plt.ylabel("P($L/a$)")
+plt.show()