More Like this

1. Local stationarity in exponential last-passage percolation

   https://doi.org/10.1007/s00440-021-01035-7  

   Balázs, Márton; Busani, Ofer; Seppäläinen, Timo (June 2021, Probability Theory and Related Fields)

   Abstract We consider point-to-point last-passage times to every vertex in a neighbourhood of size $$\delta N^{\nicefrac {2}{3}}$$ δ N 2 3 at distance N from the starting point. The increments of the last-passage times in this neighbourhood are shown to be jointly equal to their stationary versions with high probability that depends only on $$\delta $$ δ . Through this result we show that (1) the $$\text {Airy}_2$$ Airy 2 process is locally close to a Brownian motion in total variation; (2) the tree of point-to-point geodesics from every vertex in a box of side length $$\delta N^{\nicefrac {2}{3}}$$ δ N 2 3 going to a point at distance N agrees inside the box with the tree of semi-infinite geodesics going in the same direction; (3) two point-to-point geodesics started at distance $$N^{\nicefrac {2}{3}}$$ N 2 3 from each other, to a point at distance N , will not coalesce close to either endpoint on the scale N . Our main results rely on probabilistic methods only. 

   more » « less
2. Local behavior of the Eden model on graphs and tessellations of manifolds

   https://doi.org/10.1007/s41468-023-00153-6  

   Hua, Dongming_Merrick; Manin, Fedor; Queer, Tahda; Wang, Tianyi (December 2023, Journal of Applied and Computational Topology)

   Abstract The Eden Model in$${\mathbb {R}}^n$$ $R^n$constructs a blob as follows: initially a single unit hypercube is infected, and each second a hypercube adjacent to the infected ones is selected randomly and infected. Manin, Roldán, and Schweinhart investigated the topology of the Eden model in$${\mathbb {R}}^{n}$$ $R^n$by considering the possible shapes which can appear on the boundary. In particular, they give probabilistic lower bounds on the Betti numbers of the Eden model. In this paper, we prove analogous results for the Eden model on any infinite, vertex-transitive, locally finite graph: with high probability as time goes to infinity, every “possible” subgraph (with mild conditions on what “possible” means) occurs on the boundary of the Eden model at least a number of times proportional to an isoperimetric profile of the graph. Using this, we can extend the results about the topology of the Eden model to non-Euclidean spaces, such as hyperbolicn-space and universal covers of certain Riemannian manifolds. 

   more » « less
3. Convergence of the environment seen from geodesics in exponential last-passage percolation

   https://doi.org/10.4171/JEMS/1594  

   Martin, James B; Sly, Allan; Zhang, Lingfu (March 2025, Journal of the European Mathematical Society)

   A well-known question in planar first-passage percolation concerns the convergence of the empirical distribution of weights as seen along geodesics. We demonstrate this convergence for an explicit model, directed last-passage percolation on\mathbb{Z}^{2}with i.i.d. exponential weights, and provide explicit formulae for the limiting distributions, which depend on the asymptotic direction. For example, for geodesics in the direction of the diagonal, the limiting weight distribution has density(1/4+x/2+x^{2}/8)e^{-x}, and so is a mixture of Gamma(1,1), Gamma(2,1), and Gamma(3,1) distributions with weights1/4,1/2, and1/4respectively. More generally, we study the local environment as seen from vertices along geodesics (including information about the shape of the path and about the weights on and off the path in a local neighborhood). We consider finite geodesics from(0,0)ton\boldsymbol{\rho}for some vector\boldsymbol{\rho}in the first quadrant, in the limit asn\to\infty, as well as semi-infinite geodesics in direction\boldsymbol{\rho}. We show almost sure convergence of the empirical distributions of the environments along these geodesics, as well as convergence of the distributions of the environment around a typical point in these geodesics, to the same limiting distribution, for which we give an explicit description.We make extensive use of a correspondence with TASEP as seen from an isolated second-class particle for which we prove new results concerning ergodicity and convergence to equilibrium. Our analysis relies on geometric arguments involving estimates for last-passage times, available from the integrable probability literature. 

   more » « less
4. Size of nodal domains of the eigenvectors of a graph

   https://doi.org/10.1002/rsa.20925  

   Huang, Han; Rudelson, Mark (April 2020, Random Structures & Algorithms)

   Consider an eigenvector of the adjacency matrix of aG(n,p) graph. A nodal domain is a connected component of the set of vertices where this eigenvector has a constant sign. It is known that with high probability, there are exactly two nodal domains for each eigenvector corresponding to a nonleading eigenvalue. We prove that with high probability, the sizes of these nodal domains are approximately equal to each other. 

   more » « less
5. Contact graphs, boundaries, and a central limit theorem for $$\mathrm{CAT}(0)$$ cubical complexes

   https://doi.org/10.4171/GGD/775  

   Fernós, Talia; Lécureux, Jean; Mathéus, Frédéric (April 2024, Groups, Geometry, and Dynamics)

   LetXbe a nonelementary\mathrm{CAT}(0)cubical complex. We prove that ifXis essential and irreducible, then the contact graph ofX(introduced by Hagen (2014)) is unbounded and its boundary is homeomorphic to the regular boundary ofX(defined by Fernós (2018) and Kar–Sageev (2016)). Using this, we reformulate the Caprace–Sageev’s rank-rigidity theorem in terms of the action on the contact graph. LetGbe a group with a nonelementary action onX, and let (Z_{n})be a random walk corresponding to a generating probability measure onGwith finite second moment. Using this identification of the boundary of the contact graph, we prove a central limit theorem for (Z_{n}), namely that\frac{d(Z_{n} o,o)-nA}{\sqrt{n}}converges in law to a non-degenerate Gaussian distribution (A\hspace{-0.7pt}=\hspace{-0.7pt}\lim_{n\to\infty}\hspace{-0.7pt}\frac{d(Z_{n}o,o)}{n}is the drift of the random walk, ando\in Xis an arbitrary basepoint). 

   more » « less