{"620360":{"#nid":"620360","#data":{"type":"event","title":"PhD Defense by Bounghun Bock","body":[{"value":"\u003Ch2\u003EPercolation Theory: The complement of the infinite cluster \u0026amp; The acceptance profile of the invasion percolation\u003C\/h2\u003E\r\n\r\n\u003Cp\u003ESeries:\u0026nbsp;\u003C\/p\u003E\r\n\r\n\u003Cp\u003EDissertation Defense\u003C\/p\u003E\r\n\r\n\u003Cp\u003EThursday, May 2, 2019 - 1:30pm\u003C\/p\u003E\r\n\r\n\u003Cp\u003E1.5 hours (actually 80 minutes)\u003C\/p\u003E\r\n\r\n\u003Cp\u003ELocation:\u0026nbsp;\u003C\/p\u003E\r\n\r\n\u003Cp\u003ESkiles 006\u003C\/p\u003E\r\n\r\n\u003Cp\u003ESpeaker:\u0026nbsp;\u003C\/p\u003E\r\n\r\n\u003Cp\u003EBounghun Bock\u003C\/p\u003E\r\n\r\n\u003Cp\u003E,\u0026nbsp;\u0026nbsp;\u003C\/p\u003E\r\n\r\n\u003Cp\u003EGeorgia Tech\u003C\/p\u003E\r\n\r\n\u003Cp\u003E,\u0026nbsp;\u0026nbsp;\u003C\/p\u003E\r\n\r\n\u003Cp\u003E\u003Ca href=\u0022mailto:\u0022\u003Ebbock3@gatech.edu\u003C\/a\u003E\u003C\/p\u003E\r\n\r\n\u003Cp\u003EOrganizer:\u0026nbsp;\u003C\/p\u003E\r\n\r\n\u003Cp\u003E\u003Ca href=\u0022https:\/\/math.gatech.edu\/people\/bounghun-bock\u0022\u003EBounghun Bock\u003C\/a\u003E\u003C\/p\u003E\r\n\r\n\u003Cp\u003EIn independent bond percolation \u0026nbsp;with parameter p, if one removes the vertices of the infinite cluster (and incident edges), for which values of p\u0026nbsp;does the remaining graph contain an infinite cluster? Grimmett-Holroyd-Kozma used the triangle condition to show that for d \u0026gt; 18, the set of such p\u0026nbsp;contains values strictly larger than the percolation threshold pc. With the work of Fitzner-van der Hofstad, this has been reduced to d \u0026gt; 10. We reprove this result by showing that for d \u0026gt; 10\u0026nbsp;and some p\u0026gt;pc, there are infinite paths consisting of \u0026quot;shielded\u0026quot;\u0026#39; vertices --- vertices all whose adjacent edges are closed --- which must be in the complement of the infinite cluster. Using numerical values of pc, this bound can be reduced to d \u0026gt; 7. Our methods are elementary and do not require the triangle condition.\u003C\/p\u003E\r\n\r\n\u003Cp\u003EInvasion percolation is a stochastic growth model that follows a greedy algorithm. After assigning i.i.d. uniform random variables (weights) to all edges of d-dimensional space, the growth starts at the origin. At each step, we adjoin to the current cluster the edge of minimal weight from its boundary. In \u0026#39;85, Chayes-Chayes-Newman studied the \u0026quot;acceptance profile\u0026quot;\u0026#39; of the invasion: for a given p in [0,1], it is the ratio of the expected number of invaded edges until time n\u0026nbsp;with weight in [p,p+dp]\u0026nbsp;to the expected number of observed edges (those in the cluster or its boundary) with weight in the same interval. They showed that in all dimensions, the acceptance profile an(p)\u0026nbsp;converges to one for p\u0026lt;pc\u0026nbsp;and to zero for p\u0026gt;pc. In this paper, we consider an(p)\u0026nbsp;at the critical point p=pc\u0026nbsp;in two dimensions and show that it is bounded away from zero and one as n goes to infinity.\u003C\/p\u003E\r\n","summary":null,"format":"limited_html"}],"field_subtitle":"","field_summary":"","field_summary_sentence":[{"value":"Percolation Theory: The complement of the infinite cluster \u0026 The acceptance profile of the invasion percolation"}],"uid":"27707","created_gmt":"2019-04-12 14:35:37","changed_gmt":"2019-04-12 14:35:37","author":"Tatianna Richardson","boilerplate_text":"","field_publication":"","field_article_url":"","field_event_time":{"event_time_start":"2019-05-02T14:30:00-04:00","event_time_end":"2019-05-02T16:30:00-04:00","event_time_end_last":"2019-05-02T16:30:00-04:00","gmt_time_start":"2019-05-02 18:30:00","gmt_time_end":"2019-05-02 20:30:00","gmt_time_end_last":"2019-05-02 20:30:00","rrule":null,"timezone":"America\/New_York"},"extras":[],"groups":[{"id":"221981","name":"Graduate Studies"}],"categories":[],"keywords":[{"id":"100811","name":"Phd Defense"}],"core_research_areas":[],"news_room_topics":[],"event_categories":[{"id":"1788","name":"Other\/Miscellaneous"}],"invited_audience":[{"id":"78761","name":"Faculty\/Staff"},{"id":"78771","name":"Public"},{"id":"174045","name":"Graduate students"},{"id":"78751","name":"Undergraduate students"}],"affiliations":[],"classification":[],"areas_of_expertise":[],"news_and_recent_appearances":[],"phone":[],"contact":[],"email":[],"slides":[],"orientation":[],"userdata":""}}}