Phase transition in a sequential assignment problem on graphs

We study the following sequential assignment problem on a finite graph G = (V ,E). Each edge e ∈ E starts with an integer value n e ≥ 0, and we write n =∑ e∈En e. At time t, 1 ≤ t ≤ n, a uniformly random vertex v ∈ V is generated, and one of the edges f incident with v must be selected. The value of f is then decreased by 1. There is a unit final reward if the configuration (0, . . . , 0) is reached. Our main result is that there is a phase transition: as n←∞, the expected reward under the optimal policy approaches a constant c G &gt; 0 when (n e/n : e ∈ E) converges to a point in the interior of a certain convex set R G, and goes to 0 exponentially when (n e/n : e ∈ E) is bounded away from R G. We also obtain estimates in the near-critical region, that is when (n e/n : e ∈ E) lies close to ∂R G. We supply quantitative error bounds in our arguments. </p

Sandpile models

This survey is an extended version of lectures given at the Cornell Probability Summer School 2013. The fundamental facts about the Abelian sandpile model on a finite graph and its connections to related models are presented. We discuss exactly computable results via Majumdar and Dhar's method. The main ideas of Priezzhev's computation of the height probabilities in 2D are also presented, including explicit error estimates involved in passing to the limit of the infinite lattice. We also discuss various questions arising on infinite graphs, such as convergence to a sandpile measure, and stabilizability of infinite configurations

The lowest crossing in 2D critical percolation

We study the following problem for critical site percolation on the triangular lattice. Let A and B be sites on a horizontal line e separated by distance n. Consider, in the half-plane above e, the lowest occupied crossing R from the half-line left of A to the half-line right of B. We show that the probability that R has a site at distance smaller than m from AB is of order (log (n/m))^{-1}, uniformly in 1 <= m < n/2. Much of our analysis can be carried out for other two-dimensional lattices as well.Comment: 16 pages, Latex, 2 eps figures, special macros: percmac.tex. Submitted to Annals of Probabilit

Risk Analysis of Prostate Cancer in PRACTICAL Consortium--Response.

D.F. Easton was recipient of the CR-UK grant C1287/A10118. R.A. Eeles was recipient of the CR-UK grant C5047/A10692.This is the author accepted manuscript. The final version is available from the American Association for Cancer Research via http://dx.doi.org/10.1158/1055-9965.EPI-15-100

Increased level of chromosomal damage after irradiation of lymphocytes from BRCA1 mutation carriers

Deleterious mutations in the BRCA1 gene predispose women to an increased risk of breast and ovarian cancer. Many functional studies have suggested that BRCA1 has a role in DNA damage repair and failure in the DNA damage response pathway often leads to the accumulation of chromosomal aberrations. Here, we have compared normal lymphocytes with those heterozygous for a BRCA1 mutation. Short-term cultures were irradiated (8Gy) using a high dose rate and subsequently metaphases were analysed by 24-colour chromosome painting (M-FISH). We scored the chromosomal rearrangements in the metaphases from five BRCA1 mutation carriers and from five noncarrier control samples 6 days after irradiation. A significantly higher level of chromosomal damage was detected in the lymphocytes heterozygous for BRCA1 mutations compared with normal controls; the average number of aberrations per mitosis was 3.48 compared with 1.62 in controls (P=0.0001). This provides new evidence that heterozygous mutation carriers have a different response to DNA damage compared with noncarriers and that BRCA1 has a role in DNA damage surveillance. Our finding has implications for treatment and screening of BRCA1 mutation carriers using modalities that involve irradiation