The Buffon needle problem for Lévy distributed spacings and renewal theory

What is the probability that a needle dropped at random on a set of points scattered on a line segment does not fall on any of them? We compute the exact scaling expression of this hole probability when the spacings between the points are independent identically distributed random variables with a power-law distribution of index less than unity, implying that the average spacing diverges. The theoretical framework for such a setting is renewal theory, to which the present study brings a new contribution. The question posed here is also related to the study of some correlation functions of simple models of statistical physics.

The relationship between mid-infrared and sub-millimetre variability of deeply embedded protostars

ABSTRACT We study the relationship between the mid-infrared (mid-IR) and sub-millimetre (sub-mm) variability of deeply embedded protostars using the multi-epoch data from the Wide-field Infrared Survey Explorer (WISE/NEOWISE) and the ongoing James Clerk Maxwell Telescope (JCMT) Transient Survey. Our search for signs of stochastic (random) and/or secular (roughly monotonic in time) variability in a sample of 59 young stellar objects (YSOs) revealed that 35 are variable in at least one of the two surveys. This variability is dominated by secular changes. Of those objects with secular variability, 14 objects ($22{{\ \rm per\ cent}}$ of the sample) show correlated secular variability over mid-IR and sub-mm wavelengths. Variable accretion is the likely mechanism responsible for this type of variability. Fluxes of YSOs that vary in both wavelengths follow a relation of log10F4.6(t) = ηlog10F850(t) between the mid-IR and sub-mm, with η = 5.53 ± 0.29. This relationship arises from the fact that sub-mm fluxes respond to the dust temperature in the larger envelope whereas the mid-IR emissivity is more directly proportional to the accretion luminosity. The exact scaling relation, however, depends on the structure of the envelope, the importance of viscous heating in the disc, and dust opacity laws.