Combining Losing Games into a Winning Game

2015

In this paper we introduce the notion of Random Walk in Changing Environment -a random walk in which each step is performed in a different graph on the same set of vertices, or more generally, a weighted random walk on the same vertex and edge sets but with different (possibly 0) weights in each step. This is a very wide class of RW, which includes some well known types of RW as special cases (e.g. reinforced RW, true SAW). We define and explore various possible properties of such walks, and provide criteria for recurrence and transience when the underlying graph is N or a tree. We provide an example of such a process on Z 2 where conductances can only change from 1 to 2 (once for each edge) but nevertheless the walk is transient, and conjecture that such behaviour cannot happen when the weights are chosen in advance, that is, do not depend on the location of the RW. Conjecture 1.1. If Daedalus is oblivious of Theseus location, then Theseus will almost surely reach the exit (infinitely many times, if he chooses to stay in the labyrinth). In other words, Theseus' Random Walk is recurrent. See Section 7 for formal statement and further open problems.

2018

In this paper we introduce the notion of Random Walk in Changing Environment a random walk in which each step is performed in a different graph on the same set of vertices, or more generally, a weighted random walk on the same vertex and edge sets but with different (possibly 0) weights in each step. This is a very wide class of RW, which includes some well known types of RW as special cases (e.g. reinforced RW, true SAW). We define and explore various possible properties of such walks, and provide criteria for recurrence and transience when the underlying graph is N or a tree. We provide an example of such a process on Z where conductances can only change from 1 to 2 (once for each edge) but nevertheless the walk is transient, and conjecture that such behaviour cannot happen when the weights are chosen in advance, that is, do not depend on the location of the RW.

Communications in Mathematical Physics, 2000

We explain the necessary and sufficient conditions for recurrent and transient behavior of a random walk in a stationary ergodic random environment on a strip in terms of properties of a top Lyapunov exponent. This Lyapunov exponent is defined for a product of a stationary sequence of positive matrices. In the one-dimensional case this approach allows us to treat wider classes of random walks than before.

2023

Parrondo's paradox was introduced by Juan Parrondo in 1996. In game theory, this paradox is described as: A combination of losing strategies becomes a winning strategy. At first glance, this paradox is quite surprising, but we can easily explain it by using simulations and mathematical arguments. Indeed, we first consider some examples with the Parrondo's paradox and, using the software R, we simulate one of them, the coin tossing. Actually, we see that specific combinations of losing games become a winning game. Moreover, even a random combination of these two losing games leads to a winning game. Later, we introduce the major definitions and theorems over Markov chains to study our Parrondo's paradox applied to the coin tossing problem. In particular, we represent our Parrondo's game as a Markov chain and we find its stationary distribution. In that way, we exhibit that our combination of two losing games is truly a winning combination. We also deliberate possible applications of the paradox in some fields such as ecology, biology, finance or reliability theory.

Communications in Mathematical Physics, 1991

We consider random walks on Zd with transition ratesp(x, y) given by a random matrix. Ifp is a small random perturbation of the simple random walk, we show that the walk remains diffusive for almost all environmentsp ifd>2. The result also holds for a continuous time Markov process with a random drift. The corresponding path space measures converge weakly, in the scaling limit, to the Wiener process, for almost everyp.

arXiv (Cornell University), 2014

This work is motivated by the study of some two-dimensional random walks in random environment (RWRE) with transition probabilities independent of one coordinate of the walk. These are non-reversible models and can not be treated by electrical network techniques. The proof of the recurrence of such RWRE needs new estimates for quenched return probabilities of a one-dimensional recurrent RWRE. We obtained these estimates by constructing suitable valleys for the potential. They imply that k independent walkers in the same one-dimensional (recurrent) environment will meet in the origin infinitely often, for any k. We also consider direct products of one-dimensional recurrent RWRE with another RWRE or with a RW. We point out the that models involving one-dimensional recurrent RWRE are more recurrent than the corresponding models involving simple symmetric walk.

2006

We consider a model, introduced by Boldrighini, Minlos and Pellegrinotti [4, 5], of random walks in dynamical random environments on the integer lattice Z with d ≥ 1. In this model, the environment changes over time in a Markovian manner, independently across sites, while the walker uses the environment at its current location in order to make the next transition. In contrast with the cluster expansions approach of [5], we follow a probabilistic argument based on regeneration times. We prove an annealed SLLN and invariance principle for any dimension, and provide a quenched invariance principle for dimension d > 7, providing for d > 7 an alternative to the analytical approach of [5], with the added benefit that it is valid under weaker assumptions. The quenched results use, in addition to the regeneration times already mentioned, a technique introduced by Bolthausen and Sznitman [7]. AMS 2000 subject classification : 60J15,60F10,82C44,60J80.

Physical Review E, 2006

Random walk models, such as the trap model, continuous time random walks, and comb models exhibit weak ergodicity breaking, when the average waiting time is infinite. The open question is: what statistical mechanical theory replaces the canonical Boltzmann-Gibbs theory for such systems? In this manuscript a non-ergodic equilibrium concept is investigated, for a continuous time random walk model in a potential field. In particular we show that in the non-ergodic phase the distribution of the occupation time of the particle on a given lattice point, approaches U or W shaped distributions related to the arcsin law. We show that when conditions of detailed balance are applied, these distributions depend on the partition function of the problem, thus establishing a relation between the non-ergodic dynamics and canonical statistical mechanics. In the ergodic phase the distribution function of the occupation times approaches a delta function centered on the value predicted based on standard Boltzmann-Gibbs statistics. Relation of our work with single molecule experiments is briefly discussed. PACS numbers: 05.20.-y, 05.40.-a, 02.50.-r, 05.90.+m

Quantum Information Processing, 2018

A quantum walker moves on the integers with four extra degrees of freedom, performing a coin-shift operation to alter its internal state and position at discrete units of time. The time evolution is described by a unitary process. We focus on finding the limit probability law for the position of the walker and study it by means of Fourier analysis. The quantum walker exhibits both localization and a ballistic behavior. Our two results are given as limit theorems for a 2-period time-dependent walk and they describe the location of the walker after it has repeated the unitary process a large number of times. The theorems give an analytical tool to study some of the Parrondo type behavior in a quantum game which was studied by J. Rajendran and C. Benjamin by means of very nice numerical simulations [1]. With our analytical tools at hand we can easily explore the "phase space" of parameters of one of the games, similar to the winning game in their papers. We include numerical evidence that our two games, similar to theirs, exhibit a Parrondo type paradox. Keywords Quantum walk • Limit theorem • Parrondo paradox 1 Introduction Quantum walks, introduced in [2], can be considered as a counterpart of random walks and have been studied in mathematics since around 2000. With spin orientations, also known as coin states, the quantum walkers move in a discrete space in superposition. The position of a quantum walker has a limiting distribution which is very far from those of classical random walks. In

2012

Let µ 1 , . . . , µ k be d-dimensional probability measures in R d with mean 0. At each time we choose one of the measures based on the history of the process and take a step according to that measure. We give conditions for transience of such processes and also construct examples of recurrent processes of this type. In particular, in dimension 3 we give the complete picture: every walk generated by two measures is transient and there exists a recurrent walk generated by three measures.