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1. On the trace of random walks on random graphs: ON THE TRACE OF RANDOM WALKS ON RANDOM GRAPHS

   https://doi.org/10.1112/plms.12083  

   Frieze, Alan ; Krivelevich, Michael ; Michaeli, Peleg ; Peled, Ron ( April 2018 , Proceedings of the London Mathematical Society)
2. Geometry of the vacant set left by random walk on random graphs, Wright's constants, and critical random graphs with prescribed degrees

   https://doi.org/10.1002/rsa.20880  

   Bhamidi, Shankar ; Sen, Sanchayan ( July 2019 , Random Structures & Algorithms)

   We provide an explicit algorithm for sampling a uniform simple connected random graph with a given degree sequence. By products of this central result include: (1) continuum scaling limits of uniform simple connected graphs with given degree sequence and asymptotics for the number of simple connected graphs with given degree sequence under some regularity conditions, and (2) scaling limits for the metric space structure of the maximal components in the critical regime of both the configuration model and the uniform simple random graph model with prescribed degree sequence under finite third moment assumption on the degree sequence. As a substantive application we answer a question raised by Černý and Teixeira study by obtaining the metric space scaling limit of maximal components in the vacant set left by random walks on random regular graphs.

    

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3. Random sorting networks: local statistics via random matrix laws

   https://doi.org/10.1007/s00440-018-0886-1  

   Gorin, Vadim ; Rahman, Mustazee ( January 2018 , Probability Theory and Related Fields)
4. Random Dictators with a Random Referee: Constant Sample Complexity Mechanisms for Social Choice

   https://doi.org/10.1609/aaai.v33i01.33011893  

   Fain, Brandon ; Goel, Ashish ; Munagala, Kamesh ; Prabhu, Nina ( July 2019 , Proceedings of the AAAI Conference on Artificial Intelligence)

   We study social choice mechanisms in an implicit utilitarian framework with a metric constraint, where the goal is to minimize Distortion, the worst case social cost of an ordinal mechanism relative to underlying cardinal utilities. We consider two additional desiderata: Constant sample complexity and Squared Distortion. Constant sample complexity means that the mechanism (potentially randomized) only uses a constant number of ordinal queries regardless of the number of voters and alternatives. Squared Distortion is a measure of variance of the Distortion of a randomized mechanism.Our primary contribution is the first social choice mechanism with constant sample complexity and constant Squared Distortion (which also implies constant Distortion). We call the mechanism Random Referee, because it uses a random agent to compare two alternatives that are the favorites of two other random agents. We prove that the use of a comparison query is necessary: no mechanism that only elicits the top-k preferred alternatives of voters (for constant k) can have Squared Distortion that is sublinear in the number of alternatives. We also prove that unlike any top-k only mechanism, the Distortion of Random Referee meaningfully improves on benign metric spaces, using the Euclidean plane as a canonical example. Finally, among top-1 only mechanisms, we introduce Random Oligarchy. The mechanism asks just 3 queries and is essentially optimal among the class of such mechanisms with respect to Distortion.In summary, we demonstrate the surprising power of constant sample complexity mechanisms generally, and just three random voters in particular, to provide some of the best known results in the implicit utilitarian framework. 

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5. An invariance principle for one-dimensional random walks among dynamical random conductances

   https://doi.org/10.1214/19-EJP348  

   Biskup, Marek ( January 2019 , Electronic Journal of Probability)