BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Mehtaab Sawhney (MIT) DTSTART:20200817T140000Z DTEND:20200817T150000Z DTSTAMP:20260423T052331Z UID:EPC/19 DESCRIPTION:Title: Loc al limit theorems for subgraph counts\nby Mehtaab Sawhney (MIT) as par t of Extremal and probabilistic combinatorics webinar\n\n\nAbstract\nWe in troduce a general framework for studying anticoncentration and local limit theorems for random variables\, including graph statistics. Our methods i nvolve an interplay between Fourier analysis\, decoupling\, hypercontracti vity of Boolean functions\, and transference between "fixed-size" and "ind ependent" models. We also adapt a notion of "graph factors" due to Janson. \n\nAs a consequence\, we derive a local central limit theorem for connect ed subgraph counts in the Erdős-Rényi random graph G(n\,p)\, building on work of Gilmer and Kopparty and of Berkowitz. These results improve an an ticoncentration result of Fox\, Kwan\, and Sauermann and partially answers a question of Fox\, Kwan\, and Sauermann. We also derive a local limit ce ntral limit theorem for induced subgraph counts\, as long as p is bounded away from a set of "problematic" densities\, partially answering a questio n of Fox\, Kwan\, and Sauermann. We then prove these restrictions are nece ssary by exhibiting a disconnected graph for which anticoncentration for s ubgraph counts at the optimal scale fails for all constant p\, and finding a graph H for which anticoncentration for induced subgraph counts fails i n G(n\,1/2). These counterexamples resolve anticoncentration conjectures o f Fox\, Kwan\, and Sauermann in the negative.\n\nFinally\, we also examine the behavior of counts of k-term arithmetic progressions in subsets of Z/ nZ and deduce a local limit theorem wherein the behavior is Gaussian at a global scale but has nontrivial local oscillations (according to a Ramanuj an theta function). These results improve on results of and answer questio ns of the authors and Berkowitz\, and answer a question of Fox\, Kwan\, an d Sauermann.\n LOCATION:https://researchseminars.org/talk/EPC/19/ END:VEVENT END:VCALENDAR