We study random lozenge tilings of a certain shape in the plane called the Novak half-hexagon, and compute the correlation functions for this process. This model was introduced by Nordenstam and Young (2011) and has many intriguing similarities with a more well-studied model, domino tilings of the Aztec diamond. The most difficult step in the present paper is to compute the inverse of the matrix whose (i,j) entry is the binomial coefficient C(A, B_j - i) for indeterminate variables A and B_1, ..., B_n.
Metcalfe @cite_1 has developed an alternative approach to problems of this type, by developing a theory of the asymptotics of a sort of interlacing particle process. The theory currently covers a slightly different setting, in which the positions of the particles is continuous, but Metcalfe is in the process of extending his methods to the discrete setting.
@cite_1
1747067133 2078356107 2592832780 2031397756
We revisit the discrete additive and multiplicative coalescents, starting with n particles with unit mass. These cases are known to be related to some “combinatorial coalescent processes”: a time reversal of a fragmentation of Cayley trees or a parking scheme in the additive case, and the random graph process ((G(n,p))_p ) in the multiplicative case. Time being fixed, encoding these combinatorial objects in real-valued processes indexed by the line is the key to describing the asymptotic behaviour of the masses as (n + ). We propose to use the Prim order on the vertices instead of the classical breadth-first (or depth-first) traversal to encode the combinatorial coalescent processes. In the additive case, this yields interesting connections between the different representations of the process. In the multiplicative case, it allows one to answer to a stronger version of an open question of Aldous (Ann Probab 25:812–854, 1997): we prove that not only the sequence of (rescaled) masses, seen as a process indexed by the time ( ), converges in distribution to the reordered sequence of lengths of the excursions above the current minimum of a Brownian motion with parabolic drift ((B_t+ t - t^2 2, t 0) ), but we also construct a version of the standard augmented multiplicative coalescent of (Probab Theory Relat, 2013) using an additional Poisson point process. We consider a branching-selection system in R with N particles which give birth independently at rate 1 and where after each birth the leftmost particle is erased, keeping the number of particles constant. We show that, as N → ∞, the empirical measure process associated to the system converges in distribution to a deterministic measure-valued process whose densities solve a free boundary integrodifferential equation. We also show that this equation has a unique traveling wave solution traveling at speed c or no such solution depending on whether c ≥ a or c < a, where a is the asymptotic speed of the branching random walk obtained by ignoring the removal of the leftmost particles in our process. The traveling wave solutions correspond to solutions of Wiener-Hopf equations. We consider a class of branching-selection particle systems on ( R ) similar to the one considered by E. Brunet and B. Derrida in their 1997 paper “Shift in the velocity of a front due to a cutoff”. Based on numerical simulations and heuristic arguments, Brunet and Derrida showed that, as the population size N of the particle system goes to infinity, the asymptotic velocity of the system converges to a limiting value at the unexpectedly slow rate (log N)−2. In this paper, we give a rigorous mathematical proof of this fact, for the class of particle systems we consider. The proof makes use of ideas and results by R. Pemantle, and by N. Gantert, Y. Hu and Z. Shi, and relies on a comparison of the particle system with a family of N independent branching random walks killed below a linear space-time barrier. This paper examines techniques for finding falsifying trajectories of hybrid systems using an approach that we call trajectory splicing. Many formal verification techniques for hybrid systems, including flowpipe construction, can identify plausible abstract counterexamples for property violations. However, there is often a gap between the reported abstract counterexamples and the concrete system trajectories. Our approach starts with a candidate sequence of disconnected trajectory segments, each segment lying inside a discrete mode. However, such disconnected segments do not form concrete violations due to the gaps that exist between the ending state of one segment and the starting state of the subsequent segment. Therefore, trajectory splicing uses local optimization to minimize the gap between these segments, effectively splicing them together to form a concrete trajectory. We demonstrate the use of our approach for falsifying safety properties of hybrid systems using standard optimization techniques. As such, our approach is not restricted to linear systems. We compare our approach with other falsification approaches including uniform random sampling and a robustness guided falsification approach used in the tool S-Taliro. Our preliminary evaluation clearly shows the potential of our approach to search for candidate trajectory segments and use them to find concrete property violations.