 So, we derived an expression for the Fourier transform of the occupancy probability in a d dimensional space. So, in principle the problem has been solved, but it is in the conjugate space. So, we just recapitulate we showed that the Fourier transform defined as the Fourier series on the discrete lattice points of finding the probability at the nth step can be written in terms of the conjugate variables k 1, k 2, if there is a d dimensional space there will be d conjugate variables and that was given as 1 by d sigma i equal to 1 to d cos k i whole to the power n. So, this is the general solution to the symmetric random walk problem in dimensional space. The general solution to the problem now given the Fourier transform one can always obtain the occupancy probability in the real space by d dimensional Fourier inversion. So, that is we can obtain W n let us say these integers of the lattice points are denoted by m 1, m 2 etcetera in the various directions. There are d such directions and this will come as 1 by 2 pi it will be 2 pi to the power d and there will be d fold integrals minus pi to pi minus pi to pi over all k i. So, it will be d k 1 etcetera d k d then the Fourier transform will be i into k 1 m 1 plus etcetera k d m d multiplied by by definition W n tilde k 1 to k. So, this formally closes the problem, but the real insight comes only when we actually carry out this inversion for that we can quickly see that in a hypersphere or in a hyperspace the components k 1 k 2 k d etcetera can be represented as components of a vector k. So, if we define k is a vector which has components k 1 etcetera up to k d and similarly r is a vector which has components on the lattice points all integer components m 1 m d along the various dimensions. Then we can write in a compact form W n m W n r nth step the probability of being at a vector point r which merely designates all these coordinates will be 1 by 2 pi to the power d and there will be d fold integrals basically one could write d k d that is just illustrate that there will be d such integrals of e to the power minus then you can write i k dot r and W n tilde vector k just let us remember it is a k fold integral to explain the left hand side is the occupancy probability of the particle at a lattice point denoted by a vector r with the components m 1 m 2 m d up to m d at the nth step that is given as a Fourier inversion of the Fourier transformed or characteristic function we can call it of the occupancy probability in the k space. Inversion being carried out through d integrals each of them varying from minus pi to pi y minus pi to pi we saw that there is a completeness and orthogonality of d to the power k or cos k sin k functions in this space for periodic lattice problems. So, in a general hyper lattice like this. So, we are at some vector point r and we have started from some origin here 0 0 d dimensional origin. So, we random walker has executed a path possibly landing here this is one of the paths. So, we are asking for the occupancy probability at the point r via which is a superposition of several such paths with the proper weighting. Now, the we introduced the concept of the probability of return to the origin. We can ask a question if a random walker starts from the origin spans the entire hyperspace what is the probability that he could arrive at the origin at some point in time during his path. It is possible that he will arrive infinitely many times over because if he arrives once then it allows you to imagine or restart the process from that point in time and then he has to arrive once again. So, in other words we therefore, can ask a legitimate question what is the probability of return for the first time to the origin. It is somewhat closely connected to the absorber problem that we mentioned which refer to the probability of contact at a on an absorber placed at a point other than the point of starting. So, here there is no absorber. So, it is only a point of revisitation that we are looking for. Hence, we can postulate a probability probability that a random walker visits the starting point in this case origin in for the first time in k steps. For example, for in a one dimensional walk the probability that he visits in the first step is 0, but in the second step there is a probability that he would come back because in the first step he would have gone to sites minus 1 or plus 1 and. So, any side you would have a probability if it is he has gone to the plus 1 then there is a probability again of half of him visiting back to the origin from the plus 1 side and half from the minus 1 side. So, one has therefore, a finite probability of visiting the origin infinite number of steps. We denote this probability as denote by a notation f k. We have thus two concepts one there are two concepts which we should distinguish. One occupancy probability at the origin that is w let us say n 0. This merely gives the probability of finding the random walker at the nth step at origin and that finding could be a result of several returns that has happened before the nth step, but it is actually it is more like the mass or weight you associate with that point origin. So, this is a different concept and the probability of return for the first time say in n or k steps it is a k steps. These two are different concepts and to illustrate we it is useful to discuss first the concept of occupancy probability a little more with the whatever we have understand understood so far then return to the concept of the return probability in k steps. So, let us delineate a few aspects of the occupancy probability occupancy probability let us say w n 0. If we go back to the formula we have a very general expression for the occupancy probability at any site r denoted by vector r which can be obtained as a Fourier inversion of the Fourier transform of it for which we have already developed a closed form expression in terms of the sum 1 by d i equal to 1 to d cos k i to the power n. If we consider just one dimension for example, we have already a closed form solution we even obtained a closed form solution even in two dimensions. So, in general w n at the origin can be computed by just putting r equal to 0 in this expression. So, in which case it just becomes w n 0 is going to be in a d dimensional system 2 pi to the power d and there will be d fold integrals minus pi to pi d k say d fold integral then e to the power i k dot r become 0 1 because r is 0 e to the power 0 is 1. So, that will be multiplied by 1 and then w n tilde vector k in d dimensional space. So, this quantity can be understood a little better by studying the case of 1 d. For example, in 1 dimensional system we have the most general result that w n say let us say w 2 m even steps in 1 d 0 is given by 1 by 2 to the power 2 m 2 m c m. How does it come? Recall w n 1 d at m equal to 1 by 2 to the power n n c n plus m by 2 formula and since we knew that it exists only for n plus m even then only it exists and not otherwise. Now, in our case we are talking of specifically at the origin. So, we put n m equal to 0 and then it becomes 1 by 2 to the power n n c n by 2. Now, we know that since m equal to 0 n also has to be even because m is even. So, hence if put n equal to 2 m then it will be 1 by 2 to the power 2 m 2 m c m as we showed here. So, this is our starting formula from which we can derive it. We can show in fact, we had shown earlier in while discussing the properties of 1 dimensional random walk that for large m by performing a symptotic approximation to factorials using Sterling's formula w 2 m in 1 d at 0 it will go as 1 by root m. I am I am not including some of the pre factors, but as a functional behavior it has a probability will decrease very slowly in this as a square root of m. So, this is a very important property to note. And as you can understand the probability at m equal to 1 for example, w 2 in 2 steps 1 d of being in origin can put m equal to 1. So, you will have it will be 2 square and here it is 2 c 1, 2 c 1 is 2 factorial divided by 1 factorial. So, it is 2 by 4 which is half. So, in the after the first after the second step there is a probability half of the random walker being at the origin. So, we can delineate what we are saying in the asymptotics is that after sufficiently large number of steps the probabilities decrease rather slowly. We also solved similarly we also developed a solution for the occupancy probability in 2 d and if you apply that formula it actually included involved with sum it was something like n factorial divided by 4 to the power n it included a sum r equal to 0 to n. Now, if you put m equal to 0 or m 1 and m 2 both the site indices being 0 then it will be r by 2 factorial and again there will be a r by 2 factorial. So, it will be r by 2 factorial square and similarly here it will be n minus r by 2 factorial square. This follows substituting say m 1 comma m 2 equal to 0 in the expression for the solution to 2 d random walk problem. Since even in 2 dimensional case the possibility of coming back to the origin exists only in even steps for symmetric random walk we can then this whole W n equal to 2 m if we put now m here refers to steps of coming back to origin that will be it will be you can actually write it as 2 m factorial divided by 4 to the power 2 m. Now r r by 2 can be changed by replacing r equal to 2 k and then k equal to 0 to m it will be then it will be 1 by k factorial square into 1 by m minus k factorial square. This is a kind of a finite term progression we have only m plus 1 terms and we will not sum it, but this can be summed and it so turns out that this whole sum becomes 2 to the power minus 2 m 2 m factorial divided by root pi gamma m plus half divided by gamma m plus m whole cube. In fact, these kind of results are available in many handbooks and software such as Mathematica automatically sum this series and give you the results. So, this can be for example, summed in Mathematica. So, there is a closed form expression for the probability of return to the origin in a 2 dimensionals case after 2 m steps. Again we can do the asymptotics and we can see that ok we can note for example, we can assign values of m equal to 0 1 2 etcetera and we can see that w 1 0 0 that is 2 steps will become 1 4th and the next probability w 4 0 0 will turn out to be 0.141 etcetera can assign values and then obtain this. Asymptotically m tends to infinity we can use the sterling approximation for the gamma functions which are factorials we merely present the result here which says that w 2 m 2 d it will vary as 1 by m some pre factors are there I am ignoring the pre factors as m tends to infinity. So, now we have 2 results in 1 d the occupancy probability w 2 m 1 d at 0 varies as 1 by root m in 2 d same thing w 2 m 2 d 0 0 varies as 1 by m. So, let us say if m is 100 it will be varying as 1 by 100 or if we increase 100 to 400 it will become 1 4th 4 times less it was 1 by 100 and it has become 1 by 400 which is 1 by 20. So, basically it has become twice 1 by 4 or 4 times less whereas, here when m was 100 this is 1 by 10 and where m is 400 is 1 by 20. So, it is only twice less. So, the decay in one dimensional case is much slower in other words in two dimensional case there is a far more rapid dispersal tendency to disperse away from the origin is much more stronger than in one dimensional case. In other words the dimensionality of the space now has a role to play in terms of the speed of the process. One can actually plot this behavior as a function of m for 1 d and 2 d. So, if 1 d decreases like this 2 d will decrease this is for 1 d this is w 2 m at the origin and this is for 2 d there will be far more rapid decay of the probabilities of of finding the particle at the starting point at the origin in 1 d and 2 d. In fact, these probabilities will decay even faster at higher dimensions. So, with this picture of the occupancy probability we now move over to the general question of asking what is the probability of return which is a slightly different concept and what is the probability of ultimate return and in doing so we link the two concepts by a logical argument and if you know one you can calculate the other. So, they get related, but in the process we get an answer to a very important question of the what will be the probability in a general dimension or in lower dimensions is it possible to escape at all origin at all with certainty. This is called the Polytheorem which says that in 1 and 2 dimensions a random walker cannot escape revisiting the origin at some point in time he will revisit. So, if you revisit at some point in time you can restart the process and he will revisit again. So, in other words the process becomes infinitely recurrent from the property of its total time translational possibility. Thank you.