Sharp error terms for return time statistics under mixing conditions

We describe the statistics of repetition times of a string of symbols in a stochastic process. Denote by T(A) the time elapsed until the process spells the finite string A and by S(A) the number of consecutive repetitions of A. We prove that, if the length of the string grows unbondedly, (1) the distribution of T(A), when the process starts with A, is well aproximated by a certain mixture of the point measure at the origin and an exponential law, and (2) S(A) is approximately geometrically distributed. We provide sharp error terms for each of these approximations. The errors we obtain are point-wise and allow to get also approximations for all the moments of T(A) and S(A). To obtain (1) we assume that the process is phi-mixing while to obtain (2) we assume the convergence of certain contidional probabilities

Exponential distribution for the occurrence of rare patterns in Gibbsian random fields

We study the distribution of the occurrence of rare patterns in sufficiently mixing Gibbs random fields on the lattice $\mathbb{Z}^d$, $d\geq 2$. A typical example is the high temperature Ising model. This distribution is shown to converge to an exponential law as the size of the pattern diverges. Our analysis not only provides this convergence but also establishes a precise estimate of the distance between the exponential law and the distribution of the occurrence of finite patterns. A similar result holds for the repetition of a rare pattern. We apply these results to the fluctuation properties of occurrence and repetition of patterns: We prove a central limit theorem and a large deviation principle.Comment: To appear in Commun. Math. Phy

Satellites of Simulated Galaxies: survival, merging, and their relation to the dark and stellar halos

We study the population of satellite galaxies formed in a suite of N-body/gasdynamical simulations of galaxy formation in a LCDM universe. We find little spatial or kinematic bias between the dark matter and the satellite population. The velocity dispersion of the satellites is a good indicator of the virial velocity of the halo: \sigma_{sat}/V_{vir}=0.9 +/- 0.2. Applied to the Milky Way and M31 this gives V_{vir}^{MW}=109 +/- 22$ km/s and V_{vir}^{M31} = 138 +/- 35 km/s, respectively, substantially lower than the rotation speed of their disk components. The detailed kinematics of simulated satellites and dark matter are also in good agreement. By contrast, the stellar halo of the simulated galaxies is kinematically and spatially distinct from the population of surviving satellites. This is because the survival of a satellite depends on mass and on time of accretion; surviving satellites are biased toward low-mass systems that have been recently accreted by the galaxy. Our results support recent proposals for the origin of the systematic differences between stars in the Galactic halo and in Galactic satellites: the elusive ``building blocks'' of the Milky Way stellar halo were on average more massive, and were accreted (and disrupted) earlier than the population of dwarfs that has survived self-bound until the present.Comment: 13 pages, 11 figures, MNRAS in press. Accepted version with minor changes. Version with high resolution figures available at: http://www.astro.uvic.ca/~lsales/SatPapers/SatPapers.htm

Formal Verification of Security Protocol Implementations: A Survey

Automated formal verification of security protocols has been mostly focused on analyzing high-level abstract models which, however, are significantly different from real protocol implementations written in programming languages. Recently, some researchers have started investigating techniques that bring automated formal proofs closer to real implementations. This paper surveys these attempts, focusing on approaches that target the application code that implements protocol logic, rather than the libraries that implement cryptography. According to these approaches, libraries are assumed to correctly implement some models. The aim is to derive formal proofs that, under this assumption, give assurance about the application code that implements the protocol logic. The two main approaches of model extraction and code generation are presented, along with the main techniques adopted for each approac

Cosmic M\'enage \`a Trois: The Origin of Satellite Galaxies On Extreme Orbits

We examine the orbits of satellite galaxies identified in a suite of N-body/gasdynamical simulations of the formation of $L_*$ galaxies in a LCDM universe. Most satellites follow conventional orbits; after turning around, they accrete into their host halo and settle on orbits whose apocentric radii are steadily eroded by dynamical friction. However, a number of outliers are also present, we find that ~1/3 of satellites identified at $z=0$ are on unorthodox orbits, with apocenters that exceed their turnaround radii. This population of satellites on extreme orbits consists typically of the faint member of a satellite pair that has been ejected onto a highly-energetic orbit during its first approach to the primary. Since the concurrent accretion of multiple satellite systems is a defining feature of hierarchical models of galaxy formation, we speculate that this three-body ejection mechanism may be the origin of (i) some of the newly discovered high-speed satellites around M31 (such as Andromeda XIV); (ii) some of the distant fast-receding Local Group members, such as Leo I; and (iii) the oddly isolated dwarf spheroidals Cetus and Tucana in the outskirts of the Local Group. Our results suggest that care must be exercised when using the orbits of the most weakly bound satellites to place constraints on the total mass of the Local Group.Comment: 10 pages, 6 figures, MNRAS in press. Accepted version with minor changes. Version with high resolution figures available at: http://www.astro.uvic.ca/~lsales/SatPapers/SatPapers.htm

Relating two standard notions of secrecy

Two styles of definitions are usually considered to express that a security protocol preserves the confidentiality of a data s. Reachability-based secrecy means that s should never be disclosed while equivalence-based secrecy states that two executions of a protocol with distinct instances for s should be indistinguishable to an attacker. Although the second formulation ensures a higher level of security and is closer to cryptographic notions of secrecy, decidability results and automatic tools have mainly focused on the first definition so far. This paper initiates a systematic investigation of the situations where syntactic secrecy entails strong secrecy. We show that in the passive case, reachability-based secrecy actually implies equivalence-based secrecy for digital signatures, symmetric and asymmetric encryption provided that the primitives are probabilistic. For active adversaries, we provide sufficient (and rather tight) conditions on the protocol for this implication to hold.Comment: 29 pages, published in LMC