Can I get the answers for these python programming problems?Exercise $3$ The dimer covering problem $[12$ points$]$

A well studied problem in condensed matter physics is the dimer covering problem in which

dimers, meaning polymers with only two atoms, land on the surface of a solid, falling in the

spaces between the atoms on the surface and forming a grid like this:

No two dimers are allowed to overlap. The question is how many dimers we can fit in the

entire $LL$ square. The answer, in this simple case, is clearly $\frac{1}{2}LL,$ but suppose we did

not know this.

$($i$)$ Write a program to solve the problem using simulated annealing on a $2020$ lattice. The

"energy" function for the system is minus the number of dimers, so that it is minimized

when the dimers are a maximum. Start with an empty grid. The moves for the Markov

chain are as follows:

Choose two adjacent sites on the lattice at random.

If those two sites are currently occupied by a single dimer, attempt to remove the

dimer from the lattice with the appropriate probability.

If they are currently both empty, deterministically add a dimer.

Otherwise, do nothing.

Run your program and output the final energy. Plot the final configuration $($alternatively$,$

if you prefer, make an animation of the state of the system over time as the simulation

runs$).$

$($ii$)$ Try exponential cooling schedules with different time constants. A reasonable first value

to try is $=10000$ steps. For faster cooling schedules you should see that the solutions

found are poorer$-$a smaller fraction of the lattice is filled with dimers and there are

larger holes in between them$-$but for slower schedules the calculation can find quite

good, but usually not perfect, coverings of the lattice.

Hint: The main difficulty of this exercise is to distinguish if two occupied neighboring cells

are occupied by the same dimer, or by the halves of two different dimers. There are several

ways to address this. For example, giving each dimer a unique number would allow you to

keep track of which cells belong together. Alternatively, you could label the left$/$right end of

a horizontal dimer by$,$ say, $-\frac{1}{+}1$ and the bottom$/$top end of a vertical dimer by $-\frac{2}{+}2.$