BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Jan Hladky (Czech Academy of Sciences) DTSTART:20210111T140000Z DTEND:20210111T150000Z DTSTAMP:20260423T035537Z UID:EPC/42 DESCRIPTION:Title: Fli p processes on finite graphs and dynamical systems they induce on graphons \nby Jan Hladky (Czech Academy of Sciences) as part of Extremal and pr obabilistic combinatorics webinar\n\n\nAbstract\nWe introduce a class of r andom graph processes\, which we call \\emph{flip processes}. Each such pr ocess is given by a \\emph{rule} which is just a function $\\mathcal{R}:\\ mathcal{H}_k\\rightarrow \\mathcal{H}_k$ from all labelled $k$-vertex grap hs into itself ($k$ is fixed). Now\, the process starts with a given $n$-v ertex graph $G_0$. In each step\, the graph $G_i$ is obtained by sampling $k$ random vertices $v_1\,\\ldots\,v_k$ of $G_{i-1}$ and replacing the ind uced graph $G_{i-1}[v_1\,\\ldots\,v_k]$ by $\\mathcal{R}(G_{i-1}[v_1\,\\ld ots\,v_k])$. This class contains several previously studied processes incl uding the Erdos-Renyi random graph process and the random triangle removal .\n\nGiven a flip processes with a rule $\\mathcal{R}$ we construct time-i ndexed trajectories $\\Phi:\\mathcal{W}\\times [0\,\\infty)\\rightarrow\\m athcal{W}$ in the space of graphons. We prove that with high probability\, starting with a large finite graph $G_0$ which is close to a graphon $W_0 $ in the cut norm distance\, the flip process will stay in a thin sausage around the trajectory $(\\Phi(W_0\,t))_{t=0}^\\infty$ (after rescaling the time by the square of the order of the graph).\n\nThese graphon trajector ies are then studied from the perspective of dynamical systems. We prove t hat two trajectories cannot form a confluence\, give an example of a proce ss with an oscilatory trajectory\, and address the question of existence a nd stability of fixed points and periodic trajectories.\n LOCATION:https://researchseminars.org/talk/EPC/42/ END:VEVENT END:VCALENDAR