 Our next speaker will be Jose Peliguas Guerra from the University of England and the title of the talk is The Tweet Unconceited Time Emancipation Random Walks. Thanks a lot. Go ahead, Jose. Okay, I'm going to share my screen. Do you see my screen now? Not yet. Not yet. Okay, now do you see it? Nope. Now it's working, yeah. Okay, excellent. Okay, so the title of my talk is Discrete and Continuous Time Emancipation Random Walks. My full name is Jose Peliguas Guerra, but you can call me Peri, P-E-R-R-Y. Okay, so I'm from the theoretical physics group of the National Institute of Physics. It's the physics department of the main campus of the University of the Philippines. As of early this year, there were five PhD faculty in the group. In this talk, I only have two key points. The first point is there are contexts where death or mortality or evanescence of random walkers or diffusers play a major role. And the second point is many methods used to analyze systems involving immortal walkers can be modified so that they can be used to describe systems involving mortal or evanescent random walkers. Okay, so the key feature of a mortal random walk or an evanescent random walk is finite walker lifetime. In the traditional random walks we teach, there is an implicit assumption that these random walkers live forever. But in reality, there are contexts where we have to consider a finite lifetime of a random walker. And this includes photon migration in turbid biological media, detachment of molecular proteins from their polymer tracks, walker subject to radioactive decay, walker subject to photon decay, luminescence quenching, photon scattering and light absorption in tissues, scavenging reactions, stochastically moving trace hunted by a collection of predators, and target problems with mortal traps. Okay, now random walks can be discrete time random walks. You can have continuous time random walks. Now in the case of evanescent random walkers, one can make further subcategories. There are so-called step-coupled continuous time random walks, continuous time evanescent random walks, and there are step-independent continuous time evanescent random walks. In step-coupled random walks that can only occur at the moment of stepping. For example, you can have walkers that have a finite probability of being absorbed or converted into an inert species by the substrate. Every time they step from one site to another. As for the step-independent evanescent random walker, the most obvious example will be the radioactive decay of a diffusing isotope. So these isotopes can die, quote-unquote, at any moment. So that's my first point. So my next point is many methods used to analyze systems involving immortal walkers can be modified so that they can be used to describe systems involving mortal or evanescent random walkers. In the end, the rest of time, I'll only discuss one example. Specifically, I'll discuss work done with an undergraduate student. So the goal was to solve some problems involving mortal, biased, continuous time random walks. But in order to arrive at results for mortal, biased, continuous time random walks, we had to start with the traditional, with the classic random walk which we call immortal, biased discrete time random walk. And then we have to account for the finite lifetime. So that's why we have to deal with mortal bias, discrete time random walk, and then we have to account for the fact that the time between steps did not be a constant. So first let me go through the immortal, discrete time random walk in some detail. Okay, so we wish to calculate the following. The average number of visited sites as a function of time, the time-dependent return probability, the eventual return probability, and the conditional mean return time. Okay, so here's the notation. We have a bias parameter x, and so what that means is the probability of stepping to the right is r equals 1 plus x divided by 2. The probability of stepping to the left is 1 minus x divided by 2. And then we have some pretty standard notation for the probability of finding a walker at site s at time n given that it was at site s prime at time m. And then we have the notation for the probability that the random walker that started site s0 at time 0 reaches site s for the first time at step s and this is the first passage probability. And then we have delta n, which is the expected number of new sites visited on the n step. Sn is the expected number of sites, of distinct sites visited by the n step. So one can think of Sn as the sum of delta 0 plus delta 1 plus delta 2 of the delta n. Then you have r2n, the probability that the random walker returns for the first time to the origin at step 2n. Script r, the eventual return probability, and big D, the conditional mean return time. We use generating function methods. So here C is some sort of a bookkeeping parameter. And we can, and here we have the generating function for the expected number of distinct sites visited. And all the information for step 0, step 1, step 2, step n is summarized in this function. Now Sn is just the sum of the delta j's. So again delta j is the expected number of new sites visited on the j step. So we can manipulate this so that we can rewrite this expression for the generating function for the expected number of sites visited in terms of delta j. Now, how do you calculate the expected number of new sites visited at step j? Well, it turns out that it's possible to calculate the first pass, it's possible to calculate the first passage probabilities. If you sum over all the first passage probabilities on the end step, then you get delta n. And by convention delta 0 is 1. And what that means is we can get a generating function for the average number of new sites visited and express it in terms of the generating function for the first passage time, summed over all sites. Okay, now for a homogeneous random walk, it turns out that there's a relation between the generating function for the first passage probability and the generating function for the occupation probability. Okay, now if we combine all the insights we had from the four previous slides, it will turn out that one can express the generating function for the expected number of new sites visited in terms of the generating function for the occupation probability. Okay, now the probability of generating function associated with the probability of returning, and this is the p of 0 c over here, and this allows us to get the generating function for the expected number of distinct sites visited. And could you press hide on this thing that you have on the bottom? Yeah, so we can see on the screen, yeah. Okay, now is that okay? Yeah, perfect, perfect. Okay, okay. Yes, because now, so from the generating, sorry. Can you go back? Yes, okay. Another one? Another one back? Okay, yes, back? Another one back? Yeah, here, here, because we couldn't see the formula in red. Yeah, the next one. Okay, the next one. Yeah, next. One more, one more. Yeah, this formula was... Yes. This was... The previous one, the formula in red. Or you want to return to the previous one? It's the previous slide. Oh, yes. Yes, yes. So this is the generating, so this is the generating function for the expected number of distinct sites visited by step n. So if you... So in principle, if you take the Taylor series expansion of this, you can... And take the coefficients of c to the n, you get Sn. Okay, so after this, what we can do is we consider a specific random walk. So for example, I consider the bias random walk and I look for the probability and I look for the... It's possible to calculate the probability that it will return to the origin on the n-step. Okay, this is a simple binomial distribution. We get the generating function by multiplying everything by c to the 2n and take the sum and you eventually get this expression for the return probability and therefore the generating function of the return probability. Now you substitute this. So this PS0xc is the P0xc over here. And we substitute and we end up with an expression for the generating function for the survival for the number of distinct sites visited. So if we expand this in powers of c and read off the coefficients, we get the expected number of sites visited after one step, which is of course true because at the 0th step, it was at the origin. At the first step, it could have gone to the right or to the left. So you have S1 equals 2. Okay, and then you can calculate specific values for... Okay, this is the expected number of distinct sites visited by the 7th step. So take note that if there is no bias, all you get is this constant. See, 35 over 8, 65 over 6. The effect of bias is captured in the x squared, x to the 4th, and x to the 6th terms. Okay, so it's possible to calculate the first return probabilities. We have just passed the 11 minutes, so try to wrap up. Okay, so that's the first return probability. And okay, so for the immortal discrete time random walks, we have an expression for the eventual return probability. It's that one. And for the conditional mean, return time. Now for the bias, evanescent discrete time random walk. The parameters are very similar except that there's an asterisk to denote the evanescent quantities. And the main feature is to relate the evanescent quantities to the immortal quantities. So rough end is the number of surviving walkers by step and rough zero is the initial number of walkers. So you can put everything there and you can get an expression for the eventual return probability and the conditional mean return time. And it turns out that you can also get an expression for the generating function for the number of visited sites. Now, where you have an exponential decay of the number of random walkers. So, okay, you follow very similar steps. And eventually you end up with this expression for the generating function of the expected number of statistic sites visited. That's a generating function. And you get these expressions for the expected number of sites visited, say by the fifth step, fourth step, and so on. We can still calculate the first return probabilities and the conditional mean return time. Now, if we want to convert that discrete time random walk into a continuous time random walk, what we have to do is to calculate the probability that the walker stepped exactly end times by time t. And we get the Laplace transform of that and we can get the average number of sites visited by time t. As some sort of a weighted average, we take the Laplace transform of this and eventually what this says is you have almost exactly the same formula as for the discrete time random walk. But every time you see, you replace it with the Laplace transform size wiggle of you. Okay, and eventually we get this formula for the Laplace transform and we get the probability for an event less than CTRW to return to its starting point at time t. And we can calculate the eventual return probability and the conditional mean return time. If we consider an exponential weighting time distribution, it's possible to get this explicit expressions for the expected number of sites visited by time t. And what we expect as t approaches infinity. We can calculate the eventual return probability. So what we have done is to show that it's possible to calculate quantity such as the average number of visited sites, eventual return probability and conditional mean return time with just a little addition to the manipulations for the classic random walks. We have done some other related work within the group dealing with evanescent random walks. And these are the next steps. Okay, we have an idea how to do the next steps. We don't have any idea how to do this next next steps. So thank you very much for listening. Thanks a lot for this very interesting talk. So we have four minutes for questions. I have a, can I? Of course. Perry, thanks for your talk. Do you have, is there any experimental data observations that you can use your models to sort of validate and to fit to? Well, okay. I think in my second or third slide, I mentioned connections to say, connections to the diffusion of say, radioactive isotopes and so on. But I don't have data now. But the original motivation for evanescent random walks is chemical reactions where, as long as you have something that's diffusing and part of it disappears, essentially you can use this evanescent random walk as a framework. So it's been shown to work for, to fit experimental data well. I guess as well. Yeah, well, I think that was the original context. And now we're trying to make it abstract and try to see what happens if we look at things in the discrete space setting. I see. Okay. Thanks. Okay. Any more questions? Well, I have a question. Okay. Because I'm not familiar with the field. I'm from completely other field, but something I'm curious about is what is the part that was known before your investigation? I mean, what part of this? Well, I think the symmetric case was dealt with already. And in this example, what we did was to consider the effect of bias in the random walk. And of course, the big difference between the biased and the unbiased random walk is so an unbiased random walk, a bias random walk that's not biased is expected to return eventually to the origin. Random walk that's biased is not expect is not always expect does not has a probability less than one of you will expect qualitatively different. Additionally, if we talk about the unbiased random walk, the average return time turns out to be infinite. In the case of the biased random walks. Of course, the average is infinite because some of the walkers will never return. But if we consider only a conditional average, meaning we average only over all those random walkers that eventually return, you get a finite value. So what we did in addition to that is we studied how things will turn out to be different if we consider the in addition to the fact that the random walkers can die before they manage to return. Okay, thanks a lot. It was an interesting day. And I give you an applause in the name of everyone. Okay. Thank you very much. Thank you. Thank you.