Write a Python Code to simulate random walks on a 2D square lattice starting from the origin. (a) Simulate unrestricted random walks up to n=100 steps, averaging over n 2 10 walks for each n > 3. Plot (r2) as function of n (r = x + y), and extract the Flory exponent in () = A tu (t large), by an eyeball fit. (b) Simulate a 2D self-avoiding random walk (SAW). Make sure that each walk of given step-length n (polymer with given molecule number, n) has the same probability, i.e., each step direction should always be selected with probability 1/3 (except for the first step) and paths with intersections should be discarded from the average. Plot (r2) vs. n. Go to at least n-50 and use a sufficiently large nu. To extract the Flory exponent, first show analytically that (+1) (r) = 1+2v- 1 n (1) for large n. Replot your 2D SAW data using this relation and determine the value of v by an eyeball fit. Re-evaluate from part (a) with this technique. (c) Investigate fluctuations of the 2D random walk by extracting the variance, A(2), as function of n. Evaluate the exponent x in o(t) = A(r(t)) t by x the same technique as in part (b). Sketch the result as a "1-o" band around (r2) from part (a). (d) Extract the Flory exponent for a SAW on a 3D square lattice using the pro- cedure outlined in part (b). Make sure to use a sufficiently large number of walks, nu, to obtain reasonable statistics for up to n=50.