Susceptible-infected-susceptible epidemics on the complete graph and the star graph: Exact analysis

Since mean-field approximations for susceptible-infected-susceptible (SIS) epidemics do not always predict the correct scaling of the epidemic threshold of the SIS metastable regime, we propose two novel approaches: (a) an ?-SIS generalized model and (b) a modified SIS model that prevents the epidemic from dying out (i.e., without the...

Busemann functions and equilibrium measures in last passage percolation models

The interplay between two-dimensional percolation growth models and one-dimensional particle processes has been a fruitful source of interesting mathematical phenomena. In this paper we develop a connection between the construction of Busemann functions in the Hammersley last-passage percolation model with i.i.d. random weights, and the...

Semiparametric shift estimation based on the cumulated periodogram for non-regular functions

The problem of estimating the center of symmetry of a symmetric signal in Gaussian white noise is considered. The underlying nuisance function f is not assumed to be differentiable, which makes a new point of view to the problem necessary. We investigate the well-known sieve maximum likelihood estimators based on the cumulated periodogram, and...

Second class particles and cube root asymptotics for Hammersley’s process

We show that, for a stationary version of Hammersley’s process, with Poisson sources on the positive x-axis and Poisson sinks on the positive y-axis, the variance of the length of a longest weakly North–East path L(t, t) from (0, 0) to (t, t) is equal to 2E(t ? X(t))+, where X(t) is the location of a second class particle at time t . This...

Hammersley's process with sources and inks

We show that, for a stationary version of Hammersley’s process, with Poisson “sources” on the positive x-axis, and Poisson “sinks” on the positive y-axis, an isolated second-class particle, located at the origin at time zero, moves asymptotically, with probability 1, along the characteristic of a conservation equation for Hammersley’s process....