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Register a new userThe Brownian Map
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John C. Baez
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The Brownian map is a fundamental object in mathematics, in some sense a 2-dimensional analogue of Brownian motion. Here we briefly explain this object and a bit of its history.
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We study geodesics in the Brownian map $(mathcal{S},d, u)$, the random metric measure space which arises as the Gromov-Hausdorff scaling limit of uniformly random planar maps. Our results apply to all geodesics including those between exceptional points. First, we prove a strong and quantitative form of the confluence of geodesics phenomenon which states that any pair of geodesics which are sufficiently close in the Hausdorff distance must coincide with each other except near their endpoints. Then, we show that the intersection of any two geodesics minus their endpoints is connected, the number of geodesics which emanate from a single point and are disjoint except at their starting point is at most $5$, and the maximal number of geodesics which connect any pair of points is $9$. For each $1le k le 9$, we obtain the Hausdorff dimension of the pairs of points connected by exactly $k$ geodesics. For $k=7,8,9$, such pairs have dimension zero and are countably infinite. Further, we classify the (finite number of) possible configurations of geodesics between any pair of points in $mathcal{S}$, up to homeomorphism, and give a dimension upper bound for the set of endpoints in each case. Finally, we show that every geodesic can be approximated arbitrarily well and in a strong sense by a geodesic connecting $ u$-typical points. In particular, this gives an affirmative answer to a conjecture of Angel, Kolesnik, and Miermont that the geodesic frame of $mathcal{S}$, the union of all of the geodesics in $mathcal{S}$ minus their endpoints, has dimension one, the dimension of a single geodesic.
The reader is reminded of several puzzles involving randomness. These may be ill-posed, and if well-posed there is sometimes a solution that uses probabilistic intuition in a special way. Various examples are presented including the well known problem of the lost boarding pass: what is the probability that the last passenger boarding a fully booked plane sits in the assigned seat if the first passenger has occupied a randomly chosen seat? This problem, and its striking answer of $frac12$, has attracted a good deal of attention since around 2000. We review elementary solutions to this, and to the more general problem of finding the probability the $m$th passenger sits in the assigned seat when in the presence of some number $k$ of passengers with lost boarding passes. A simple proof is presented of the independence of the occupancy status of different seats, and a connection to the Poisson--Dirichlet distribution is mentioned.
We consider the Brownian interlacements model in Euclidean space, introduced by A.S. Sznitman in cite{sznitman2013scaling}. We give estimates for the asymptotics of the visibility in the vacant set. We also consider visibility inside the vacant set of the Brownian excursion process in the unit disc and show that it undergoes a phase transition regarding visibility to infinity as in cite{benjamini2009visibility}. Additionally, we determine the critical value and that there is no visibility to infinity at the critical intensity.
We survey the published work of Harry Kesten in probability theory, with emphasis on his contributions to random walks, branching processes, percolation, and related topics. A complete bibliography is included of his publications.
We examine two aspects of the mathematical basis for two-tier voting systems, such as that of the Council of the European Union. These aspects concern the use of square-root weights and the choice of quota. Square-root weights originate in the Penrose square-root system, which assumes that votes are cast independently and uniformly at random, and is based around the concept of equality of influence of the voters across the Union. There are (at least) two distinct definitions of influence in current use in probability theory, namely, absolute and conditional influence. These are in agreement when the underlying random variables are independent, but not generally. We review their possible implications for two-tier voting systems, especially in the context of the so-called collective bias model. We show that the two square-root laws invoked by Penrose are unified through the use of conditional influence. In an elaboration of the square-root system, Slomczynski and Zyczkowski have proposed an exact value for the quota $q=q^*$ to be achieved in a successful vote of a two-tier system, and they have presented numerical and theoretical evidence in its support. We indicate some numerical and mathematical issues arising in the use of a Gaussian (or normal) approximation in this context, and we propose that other values of $q$ may be as good if not better than $q^*$. We discuss certain aspects of the relationship between theoreticians and politicians in the design of a two-tier voting system, and we reach the conclusion that the choice of quota in the square-root system is an issue for politicians informed by theory.
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