L = {x such that x is an acceptable Riemann Hypothesis proof} is in NP and therefore, using a reduction, we can come up with a circuit C that takes as input x and outputs 1 if x is a proof for the Riemann Hypothesis or 0 otherwise. Our goal now is to design a pair of PPT machines (Enc,Dec) such that: 1. Enc(C,m) takes as input the circuit C and m ∈ {0, 1} and outputs a ciphertext e ∈ {0, 1}∗. 2...

Grimmett, Kesten and Zhang (1993) showed that for d ≥ 3, simple random walk on the infinite cluster C∞(Z, p) of supercritical percolation on Zd is a.s. transient. Their result is equivalent to the existence of a nonzero flow f on the infinite cluster such that the 2–energy ∑ e f(e) 2 is finite. Here we sharpen this result, and show that if d ≥ 3 and p > pc(Z), then C∞(Z, p) supports a nonzero f...

In this paper we study the absolute continuity of self-similar measures defined by iterated function systems (IFS) whose contraction ratios are not uniform. We introduce a transversality condition for a multi-parameter family of IFS and study the absolute continuity of the corresponding self-similar measures. Our study is a natural extension of the study of Bernoulli convolutions by Solomyak, P...

Benjamini, Pemantle, and Peres constructed nearest neighbor processes which have predictability profiles that decay faster than that of the simple random walk. Häggström and Mossel found processes with even faster decaying predictability profiles. We prove that rate of decay achieved by Häggström and Mossel is optimal.

A well-known result of Benjamini, Lyons, Peres, and Schramm states that if G is a finitely generated Cayley graph group Γ , then amenable only admits -invariant random spanning tree with at most two ends. We show this equivalent to the existence double ray in power .

We present a Monte Carlo algorithm for efficiently finding near optimal moves and bids in the game of Bidding Hex. The algorithm is based on the recent solution of Random-Turn Hex by Peres, Schramm, Sheffield, and Wilson together with Richman’s work connecting random-turn games to bidding games.

We show how the evolving set methodology of Morris and Peres can be used to show Cheeger inequalities for bounding the spectral gap of a finite Markov kernel. This leads to sharp versions of several previous Cheeger inequalities, including ones involving edge-expansion, vertex-expansion, and mixtures of both. A bound on the smallest eigenvalue also follows.

A hybrid formalism is proposed for interacting classical and quantum sytems. This formalism is mathematically consistent and reduces to standard classical and quantum mechanics in the case of no interaction. However, in the presence of interaction, the correspondence principle is violated. PACS numbers: 03.65.Bz, 03.65.Sq ∗ Electronic mail: [email protected], [email protected]

Considering the optimal quantum cloning transformations for equatorial qubits, the quantum cloning networks for equatorial qubits consisting of quantum gates are presented. The copied equatorial qubits are separable by using Peres-Horodecki criterion. The phase-covariant optimal quantum triplicators are given. PACS: 03.67.-a, 03.65.Bz, 89.70.+c.