These notes provide an elementary and self-contained introduction to branching random walks. Section 1 gives a brief overview of Galton–Watson trees, whereas Section 2 presents the classical law of large numbers for branching random walks. These two short sections are not exactly indispensable, but they introduce the idea of using size-biased trees, thus giving motivations and an avant-goût to ...

Lyons, Pemantle and Peres asked whether the asymptotic lower speed in an innnite tree is bounded by the asymptotic speed in the regular tree with the same average number of branches. In the more general setting of random walks on graphs, we establish a bound on the expected value of the exit time from a vertex set in terms of the size and distance from the origin of its boundary, and prove this...

Lyons, Pemantle and Peres asked whether the asymptotic lower speed in an innnite tree is bounded by the asymptotic speed in the regular tree with the same average number of branches. In the more general setting of random walks on graphs, we establish a bound on the expected value of the exit time from a vertex set in terms of the size and distance from the origin of its boundary, and prove this...

Low Power dissipation and smaller area are one of the important factors while designing multipliers for digital circuits. As multipliers used in digital circuits dissipate large amount of heat whenever there is a transition of bits. Reversible Logic has emerged as a promising technology in reducing power dissipation. It has application in various fields such as low power VLSI, Quantum computing...

Due to the importance of quantum entangled states in quantum information and computation [1, 2, 3], much effort has been done recently towards an operational characterization of separable states [4, 5, 6]. The manifestations of mixed-state entanglement can be very subtle [7]. Till now there is no general efficient criterion in judging the separability. The Bell inequalities [8], Peres PPT crite...

Abstract. It is shown that the f -divergence between two probability measures P and R equals the supremum of the same f -divergence computed over all finite measurable partitions of the original space, thus generalizing results previously proved by Gel’fand and Yaglom and by Peres for the Information Divergence and more recently by Dukkipati, Bhatnagar and Murty for the Tsallis’ and Rényi’s div...

It is shown that the “retrodiction paradox” recently introduced by Peres [Phys. Lett. A 194 ( 1994) 211 arises not because of the fallacy of the time-symmetric approach as he claimed, but due to an inappropriate use of retrodiction. In a recent Letter [ 11 Peres claimed that our timesymmetric approach to quantum theory [2,3], in which retrodictions and predictions are on an equal footing, leads...

We prove the dimensional reduction conjecture of Fey, Levine, and Peres (2010) on hypercube. The proof shows that reduction, symmetry, regularity Abelian sandpile persist during parallel toppling process. This stronger result verifies empirical observations first documented by Liu, Kaplan, Gray (1990).