Convergence to the Tracy-Widom distribution for longest paths in a directed random graph

We consider a directed graph on the 2-dimensional integer lattice, placing a directed edge from vertex $(i_1,i_2)$ to $(j_1,j_2)$, whenever $i_1 \le j_1$, $i_2 \le j_2$, with probability $p$, independently for each such pair of vertices. Let $L_{n,m}$ denote the maximum length of all paths contained in an $n \times m$ rectangle. We show that there is a positive exponent $a$, such that, if $m/n^a \to 1$, as $n \to \infty$, then a properly centered/rescaled version of $L_{n,m}$ converges weakly to the Tracy-Widom distribution. A generalization to graphs with non-constant probabilities is also discussed.Comment: 20 pages, 2 figure

Nomenclatural-syntaxonomic revision of the Ass.."Drypeto-Linarietum parviflorae Horvatić et Domac 1957" (Peltarion Alliaceae)

U radu je izvršena nomenklaturna i sintaksonomska revizija asocijacije "Drypeto-Linarietum parviflorae", koju su opisali Horvatić i Domac iz Biokova u Hrvatskoj. Determinacijom na temelju herbarskog materijala je ustanovljeno da "Linaria simplex (Willd.) DC"., odnosno, "L. parviflora (Jacq.) Hal." kako ju je odredio Domac pripada u stvari vrsti Linaria microsepala A. Kerner, pa je i ime same asocijacije moralo biti promijenjeno u Linario microsepalae-Drypetum jacquinianae Trinajstić, nom. et comb. nov. (hoc loco). Navedenu asocijaciju treba, u opsegu sveze Peltarion alliaceae podrediti balkanskom redu Drypetalia spinosae i razredu Drypetea spinosae, a ne alpskom redu Thlaspietalia rotundifolii i razredu Thlaspietea rotundifolii.In the paper, the nomenclatural and syntaxonomic revision of the association "Drypeto-Linarietum parviflorae", from Biokovo in Croatia, describet by Horvatić and Domac was made. By determination based on the herbarium material it was found that "Linaria simplex (Willd.) DC." viz. "Linaria parviflora (Jacq.) Hal." as determined by Domac beloned in fact to the species Linaria microsepala A. Kerner, and therefore, the name of the association had to be changed to Linario microsepalae-Drypetum jacquinianae Trinajstić, nom. et comb. nov. (hoc loco). This association should, within the alliance Peltarion alliaceae, be subordinated to the Balkanic order Drypetalia spinosae and the class Drypetea spinosae, and not to the Alpine order Thlaspietalia rotundifolii and the class Thlaspietea rotundifolii