 Let us continue with the unbiased one dimensional random walk. A pictorial representation of the whole process is sort of shown here. A lattice along the x axis, we have just shown about plus 4 sides, 4 sides to the positive side and 4 sides to the negative side, but the lattice actually extends from minus infinity to infinity. So, a walker starts from the origin then he could take many paths in the course of n steps. So, steps are represented by the y axis. So, in the course of let us say 4 steps he can pass through let us say the site 1 and back to site 0 and then again to site 1 and then to site 2 as path 1 or he could have opted for minus side in the first jump minus 1 and then again to minus 2 come to 3 and then to come to again site 2 in the third step and then to back to site 3 or so in the third step. So, 4th step. So, one could choose many paths and many a times one can come to the same point which we are not shown here, he can cross the starting point itself and paths can meet. So, a particular realization has to be repeated several times and several realizations have to be obtained to find some average behavior. So, the occupancy probability is the quantity of interest one will then ask what is the probability that after n steps a person occupy or a random walker occupies a particular state say m equal to minus 2. What is the probability that after 4 steps he is at minus 2 or for that matter after 10 steps he is at minus 2. One thing we know that if it takes n steps he should be lying somewhere between minus n to plus n. So, m as is location and index must lie between minus n to plus n that is one information and second information that we can see is that whenever he takes a step if the step is odd he will be at an odd side. So, in the first step for example, he could be in site 1 or minus 1 in the second step he could be at the site 0 or site 2 there is he cannot be at site 1. So, this is quite obvious. So, in other words m is odd or even depending on whether n we solve these points. So, with this we yesterday arrived at the occupancy probability expression which we denoted by w subscript n the probability that a random walker is at position or site or state m at the nth step is given by 1 by 2 to the power n n factorial divided by n minus m by 2 factorial multiplied by n plus m by 2 factorial. So, very simple expression which we can used use for evaluating the occupancy probability. The problem is completely solved all the information that we require now are contained in this probability distribution. We can calculate the mean we can calculate variances we can obtain the plots of locations we can examine how it behaves as the steps increases all these information can be obtained from w n n. To familiarize ourselves we can actually do some some kind of an estimation. For example, let us say that how exactly he is going to be distributed at various steps at the first few steps will give us an idea. Say let us say first n equal to 0 the the walker will be at site 0 only is at the origin. So, n equal to 1 in the first step he will be at location minus 1 ok let us put this way this is the this is the state or site index and this is the probability index the probability will be unity at site 0 and it will be 0 elsewhere. So, at the first step for example, he will be either at minus 1 or at plus 1 with the probability half and half here this total probability is always conserved. Now, let us go on step ahead let us look at what happens at n equal to 2. Now, we know that evenness and oddness information and symmetry information. So, it makes it easier. So, he could be at minus 2 he could be at 0 he could be at plus 2. So, easy to see that if you put here n equal to 2 at m equal to 0 his probability will be half because it should be symmetric. So, it should be one fourth and one fourth here one could have easily written down this from the fact that twice of this plus this equal to 1. So, once we know one of them you could write down the others for n equal to 3 similarly we can estimate. Now, the sites are going to be minus 3 minus 1 plus 1 and plus 3 these many options. If you go back to the expression here now if you put n equal to 3 3 factorial divided by 3 minus m by 2 and 3 plus m by 2 we again start putting values m equal to 0 he cannot exist because we know that odd cases only m also should be odd. So, the first thing you will do is to put m equal to 1 and then we are going to get 1 it is going to be 3 by 8. So, minus 1 also it should be symmetrical. So, minus 3 by 8. So, then we are left with 1 by 8 here and 1 by 8 here. So, the total will add to 1 sorry it is always plus probabilities are always plus. So, 1 plus 3 4 plus 3 7 plus 1 8. So, it is conserved. So, likewise we can continually see the branching that is taking place more and more sites are covered and probability eventually gets distributed or particles get distributed with various probability at different sites. We can for example, the same thing can also be plotted just this is not exactly to the scale, but just to give you a feeling 1 2 3. Now, we can plot W n m and the y axis site on the x axis similarly here minus 1 minus 2 minus 3. So, for each depending on the number of steps you will have different distributions. So, one could have for example, distribution for an even number of steps if you want then for even steps it will be always present only at even places. So, like it will be exist at 0 and then at 2 here. So, it will exist at 0 2 and minus 2 then of course, minus 4 etcetera. So, you will get probability masses assigned to each of this sites. So, and as n increases the distribution will continue to evolve. So, this is the way we can represent. An important information that we can obtain is an question what is the probability of finding the walker at the origin? Finding the walker random walker unbiased random walker at m equal to 0 that is origin where is started. This is a very interesting question it is unique to random walk. So, that that question has a definite answer. We go to the expression formula that we have and then we see that when we put m equal to 0 we will obtain W n 0 will be equal to when you put m equal to 0 it will this of course, will be remain as such and it is n factorial here. But when m equal to 0 both those terms will simply become n by 2 factorials. So, it will be n by 2 factorial whole square you can. So, this probability of course, will exist only when n is even for all n equal to r cases he will be he cannot visit any one site. So, that is a very important difference between a discrete random walk and a continuous random walk. Later of course, one will move to a continuous random walk where it will be averaged over these odd and even case probabilities that is the different perception all together. But if you strictly go by discrete random walk we have the situation that entire probability disappears and reappears as the walker takes different steps. We have seen earlier that we can also estimate the moments of the distribution that is most important them are the mean and the variance or standard deviation. Of course, mean is always we right now we have the information we can always obtain it by summing over all states minus n to n with respect to m W n m this will be the definition of mean or m square average if we want that will also come as sum over the same occupancy probabilities of the quantity m square. This is of course, one way to do it one can always square and sum, but more elegant and a more simple way is to use the generating function itself which we have derived and we have seen given a generating function there is the two relationships we have seen the one is that the mean m bar will always be the first derivative of the generating function z where z equal to 1 or simply written as g prime 1. Similarly, we have seen that the variance sigma square will involve g double prime 1 plus g prime 1 minus g prime square 1 then subscript it if you want the variance at nth step then g will be subscripted with n steps, but these are general definitions. We have seen it earlier when we work with general properties of the generating function. So, we will use in our case for 1 d random walk qualify it as unbiased or symmetric. We have seen the nth step generating function g n z was seen to be z plus 1 by z to the power n 1 by 2 to the power n. So, we can easily work now take its derivative for example, g n n prime z will be the same 1 by 2 to the power n n z plus 1 by z to the power n minus 1 into 1 minus 1 by z square. So, if you want to evaluate at z equal to 1 then we have to evaluate this also at z equal to 1 and as we can see when z equal to 1 this particular term will go to 0 other terms will be regular. So, this will tend to 0. In fact, at z equal to 1 it will be exactly equal to 0. This is as expected which means that m bar equal to 0 that is why it is called symmetric random walk which means on an average ultimately if you look at his position on an average will be 0. He will be essentially where he is that is the perception one will have and we always think that doing random going randomly both left and right is of no use because we just remain where we are. This is the way we commonly perceive a random walk. So, always give more emphasis to a purposive walk or something what we say. However, the interesting part is it is not just the mean which decides its extent of utility in terms of displacements and movements, but it is a variance around the that mean. So, if you to a second derivative for example, it will involve all one more derivatives g and double prime z if you do it will have for example, you can write it as n g and z z square minus 1 divided by z square plus 1 plus n g n z 4 z divided by z square plus 1 whole square and when you put z equal to 1 this expression you can easily verify yourself by differentiating and this will become n because this term will go to 0 and z equal to 1 this will be 2 square 4 and that 4 will cancel g and 1 is 1. So, you are left with n. Hence we have an important property that sigma square since the mean is 0 sigma square will also be n because it will be n plus g prime is 0 both the terms will be 0 it will be just n. Hence the standard deviation is proportional to root n we should remember that earlier we had obtained this result without actually going to a complete without having a full knowledge of the distribution by methodology adopted for deriving central limit theorem. So, in so far as the standard deviation is concerned one can obtain that without having to derive the random work equation or find it is a solution. We are only reconforming that yes that kind of an analysis is correct it is consistent with the detailed occupancy probability that we have evolved. The implication of this is as the walker proceeds and if I have to show how what it means is. So, if one has let us say n equal to 1 one has probabilities at this plus at minus 1 and plus 1 and as an n equal to 3 let us go through only odd positions 1 and let us say 2 and 3 here minus 2 and minus 3. Now the distribution at minus 3 for example, could be another if you show it like this at minus 3 and 3 it would be broader. So, basically this broadening of the distribution is all that is indicated by saying that the standard deviation is increases monotonically with the step size. This distribution becomes broader and broader and of course, one you will you will bring forward eventually a result which we are going to prove soon that in the very large n limit this all this gets approximated to a Gaussian thereby implying that originally whatever was a narrower Gaussian. So, this is say n equal to 100 and this let us say will be n equal to 200 just to indicate. So, it becomes broader and broader as n increases that is the implication of the result. So, let us go a step further and try to understand how does this distribution evolve for large n. So, we call it as n tends to infinity limit of the random walk distribution. This is also called the method of asymptotics. We have seen our derivation for showing how the Poisson distribution goes over to a Gaussian distribution we approached we applied a method very similar method we will apply here. However, every time we carry out an asymptotic approximation to an exact solution one must be able to identify the correct variable about which I want to obtain my final expression. A symptotic approximation is not a very fixed thing depending on the choice of variables or the choice of the groups one may acquire or one may obtain the right solution or one may end up with getting an improper solution or incorrect solution. So, we must here identify very carefully the variable choices and that is why we will work this out in detail. Let us start with the exact expression for W n m is 1 by 2 to the power n n factorial divided by n plus m by 2 factorial into n minus m by 2 factorial. We will write this as numerator divided by denominator. So, first let us look at the numerator we will simply denote the numerator as a num equal to n factorial. Now, we use a sterling approximation for n greater much greater larger than 1 we do not take the full limit of to going to infinity, but large n then we know that n factorial is n to the power n e to the power minus n root 2 pi n. So, we keep this in mind let us call it as 1. Now, let us come to the denominator. So, this is numerator the denominator is defined as 2 to the power n n plus m by 2 factorial into n minus m by 2 factorial. So, we will apply the sterling approximation to this very carefully we will have to combine terms. So, we will come write terms in the form of the type n to the power n first e to the power minus n next and square root of 2 pi n the third part. So, this will have to be written 2 to the power n will remain as 2 to the power n and here I will use the symbol nearly equal to asymptotically it is going to go. So, this will be n plus m by 2 to the power n plus m by 2 this is equivalent to n to the power n. The second term will similarly give you n minus m by 2 divided by n minus m by 2. Now, about now come to the exponential we are writing it. So, exponential form now for the first term will be e to the power minus n plus m by 2 and for the second term the same exponential will be e to the power minus n minus m by 2 and the last one is going to be square root of there will be 2 pi and 2 pi. So, it is going to be 4 pi square and it is going to be n plus m and n minus m. So, it can write straight away as n square there will be of course, it is n plus m by 2 and n minus m by 2. So, that 4 will come here. So, it will cancel actually. So, let us very very carefully first the difficult part here is the first 2 terms. So, let us finish off the so called low hanging fruits the last terms. So, they can easily be combined to give from now on I will again use equal to. So, it is going to be 2 to the power n here. Exponentials you will see that e to the power minus m by 2 and e to the power plus m by 2 will cancel e to the power minus n by 2 and e to the power minus n by 2 will combine to give you e to the power minus n and the square root term will be simply square root of pi square into n square minus n square and the back to the other terms we will have to write it as we will take out this 2 or maybe we will do it in the next step n plus m by 2 to the power n plus m by 2 and n minus m by 2 n minus m by 2. Now, it is time to combine the last terms we can take out the 2 to the power first we can see that it will be 2 to the power n by 2 plus m by 2 and 2 to the power n by 2 minus m by 2. So, the m contribution will vanish. So, that will then kill the 2 to the power n that is in the numerator here. So, we are just discussing entirely now the denominator and here the pi will come out. So, we can keep the pi out and this will be n square minus m square and now this is n plus m to the power n by 2 n minus m to the power n by 2. So, we can visualize it as n plus m into n minus m to the power n by 2 and second are going to be n plus m to the power m by 2 n minus m to the power minus n by 2. So, this will be n plus m divided by n minus m to the power m by 2. We will continue with this shortly. Thank you.