Sarai Hernandez-Torres

Papers

1. Scaling limits of the three-dimensional uniform spanning tree and the associated random walk,
   with Omer Angel, David Croydon and Daisuke Shiraishi. Annals of Probability, to appear.
   [ arXiv | pdf | abstract | BibTeX ]

   We show that the law of the three-dimensional uniform spanning tree (UST) is tight under rescaling in a space whose elements are measured, rooted real trees, continuously embedded into Euclidean space. We also establish that the relevant laws actually converge along a particular scaling sequence. The techniques that we use to establish these results are further applied to obtain various properties of the intrinsic metric and measure of any limiting space, including showing that the Hausdorff dimension of such is given by 3/β, where β≈1.624… is the growth exponent of three-dimensional loop-erased random walk. Additionally, we study the random walk on the three-dimensional uniform spanning tree, deriving its walk dimension (with respect to both the intrinsic and Euclidean metric) and its spectral dimension, demonstrating the tightness of its annealed law under rescaling, and deducing heat kernel estimates for any diffusion that arises as a scaling limit.

   @article{/2003.09055, Author = {Omer Angel and David A. Croydon and Sarai Hernandez-Torres and Daisuke Shiraishi}, Title = {Scaling limits of the three-dimensional uniform spanning tree and associated random walk}, Year = {2020}, Note = {Preprint. Available at \url{https://arxiv.org/abs/2003.09055}} }

2. The number of spanning clusters of the uniform spanning tree in three dimensions,
   with Omer Angel, David Croydon and Daisuke Shiraishi. Advanced Studies in Pure Mathematics, to appear.
   [ arXiv | pdf | abstract | BibTeX ]

   Let U_δ be the uniform spanning tree on δZ^3. A spanning cluster of U_δ is a connected component of the restriction of U_δ to the unit cube [0, 1]^3 that connects the left face {0} × [0, 1]^2 to the right face {1}×[0, 1]^2. In this note, we will prove that the number of the spanning clusters is tight as δ → 0, which resolves an open question raised by Benjamini in [Benjamini, Large scale degrees and the number of spanning clusters for the uniform spanning tree].

   @article{2003.04548, Author = {Omer Angel and David A. Croydon and Sarai Hernandez-Torres and Daisuke Shiraishi}, Title = {The number of spanning clusters of the uniform spanning tree in three dimensions}, Year = {2020}, Note = {To appear in Adv. Stud. Pure Math. Proceedings of “The 12th Mathematical Society of Japan, Seasonal Institute (MSJ-SI) Stochastic Analysis, Random Fields and Integrable Probability.”} }

3. Chase-escape with death on trees,
   with Erin Beckman, Keisha Cook, Nicole Eikmeier and Matthew Junge. Annals of Probability, to appear.
   [ arXiv | pdf | abstract | BibTeX ]

   Chase-escape is a competitive growth process in which red particles spread to adjacent uncolored sites while blue particles overtake adjacent red particles. This can be thought of as prey escaping from pursuing predators. On d-ary trees, we introduce the modification that red particles die and describe the phase diagram for red and blue particle survival as the death rate is varied. Our analysis includes the behavior at criticality, which is different than what occurs in the process without death. Many of our results rely on novel connections to weighted Catalan numbers and analytic combinatorics.

   @article{1909.01722, Author = {Erin Beckman and Keisha Cook and Nicole Eikmeier and Sarai Hernandez-Torres and Matthew Junge}, Title = {Chase-escape with death on trees}, Year = {2019}, Note = {Preprint. Available at \url{https://arxiv.org/abs/1909.01722}} }

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Online talks

• Atrápame si puedes: un sistema de partículas interactivas. Coloquio ex alumnos CIMAT-DEMAT, February 2021.
• Three-dimensional uniform spanning trees. Joint Israeli Probability Seminar, January 2021.
• Scaling limits of uniform spanning trees in three dimensions. UBC Probability Seminar, September 2020.
• La geometría de los fenómenos críticos. Coloquio Oaxaqueño de Matemáticas, October 2020.

Talks by coauthors

• David Croydon: Scaling limits of the two- and three-dimensional USTs. Oxford Discrete Mathematics and Probability Seminar, October 2020.

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Theses

• A macroscopic view of two discrete random models,
  PhD Thesis. Supervisors: Omer Angel and Martin Barlow.
  [ UBC library | abstract | | slides for the oral defence | ]

  This thesis investigates the large-scale behaviour emerging in two discrete models: the uniform spanning tree on ℤ³ and the chase-escape with death process. We consider the uniform spanning tree (UST) on ℤ³ as a measured, rooted real tree, continuously embedded into Euclidean space. The main result is on the existence of sub-sequential scaling limits and convergence under dyadic scalings. We study properties of the intrinsic distance and the measure of the sub-sequential scaling limits, and the behaviour of the random walk on the UST. An application of Wilson’s algorithm, used in the study of scaling limits, is also instrumental in a related problem. We show that the number of spanning clusters of the three-dimensional UST is tight under scalings of the lattice. Chase-escape is a competitive growth process in which red particles spread to adjacent uncoloured sites while blue particles overtake adjacent red particles. We propose a variant of the chase-escape process called chase-escape with death (CED). When the underlying graph of CED is a d-ary tree, we show the existence of critical parameters and characterize the phase transitions.

• Introduccion a los espacios de Bergman (Introduction to Bergman spaces),
  Tesis de Licenciatura (Undergraduate thesis, in Spanish). Director de tesis (thesis advisor): Fernando Galaz Fontes.
  [ pdf | resumen (abstract) | ]

  Esta tesis de licenciatura expone los primeros resultados que se presentan en la teoría de los espacios de Bergman, y que son fundamentales para el desarrollo posterior. Su objetivo principal es introducir el tema de manera sencilla, limitándose a demostraciones elementales y una presentación autocontenida. De esta forma, buscamos que el trabajo sea accesible para un estudiante que haya tomado cursos introductorios de análisis funcional, teoría de la medida y variable compleja.