Gambler's Ruin: The Limiting Distribution of Points Visited Exactly Once of a Simple Random Walk Up to Time of Exit and the Influencer Voter Model

Abstract

In this thesis we analyze two stochastic processes. We first consider the simple symmetric random walk on the domain $ [0,N]\subset\mathbb{Z} $ with a particular interest in the points visited exactly once by the random walk up to time of exit, denoted by $ R^1_N $. Specifically, we are interested in determine a scaling limit of $ R^1_N $ as $ N\uparrow \infty $. Making use of the simple geometry, we give a combinatorial argument and use the classic Gambler's ruin identities of the symmetric random walk to determine the moments of $ R^1_N $. We then find that $ R^1_N/\log(N) $ converges weakly to the exponential distribution $ \text{Exp}(2) $. \\ The second process we consider is a variant of the classic voter model on the complete graph with $ N $ nodes, which we call the influencer model. By allowing a subset of nodes, called the influencers, to be more likely to have their opinion adopted by a neighbor we generalize the classical voter model. We establish a weak law of large numbers via the fluid limit for the influencer model and find that the average trajectory of the process is deterministic and is the solution to a set of ordinary differential equations. Let $ \rho_R(t) $ be the density of voter of opinion 1 in the regular nodes at time $ t $ and let $ \rho_I(t) $ be the density of voters of opinion 1 in the influencer nodes at time $ t $. We then determine the probability and time to consensus using martingale methods with the martingale $ Y(t)=p\rho_I(t)+q\rho_R(t) $. We derive that the probability of reaching consensus on 1 is equal to $ p\rho_I(0)+q\rho_R(0) $. For the expected time to consensus we arrive at a simple upper bound and then we also give a heuristic argument based on the results of the fluid limit section that gives an exact expectation. For our upper bound derived from using the martingale, we get that the expected time to consensus $ E[\tau] =O(N)$ with upper bound \[ E[\tau] \leq \frac{N}{C}\mathbb{E}[Y^2_0].\] In the heuristic section we find that \[\mathbb{E}[\tau] = \left[\frac{4RI}{N}\left((1-\omega) \log\left(\frac{1}{1-\omega}\right) +\omega \log\left(\frac{1}{\omega}\right)\right) \right] \] where $ \omega = p\rho_R(0) + q\rho_I(0) $ and $ R,I $ are equal to the number of regular and influencer nodes, respectively.