### About

I am a PhD candidate working with the
probability group
in the
Department of Mathematics
at the University of British Columbia.
My supervisors are Omer Angel and
Martin Barlow.
I am supported by the Mexican National Council for Science and Technology
(Consejo Nacional de Ciencia y Tecnología, CONACYT ).

Prior to this, I completed my M.Sc. in Mathematics at UBC under the supervision of
Martin Barlow. I got my B.Sc. in Mathematics at
University of Guanajuato / CIMAT in Mexico.

Here is my CV

### Research interests

My research is on discrete probability. I study stochastic models defined over combinatorial structures,
such as random walks, random trees, and competitive growth processes. My work describes the large-scale
behavior of discrete random processes and their rigorous connections to continuous models through the
scaling limit.

The Continuum Random Tree (aka CRT or Brownian CRT) is my favourite random object.

### Publications and e-prints

*The number of spanning clusters of the uniform spanning tree in three dimensions*,
with
Omer Angel,
David Croydon and
Daisuke Shiraishi.
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.”

[ 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 = {Preprint. Available at \url{https://arxiv.org/abs/2003.04548}},
}

*Scaling limit of the three-dimensional uniform spanning tree and the associated random walk*,
with
Omer Angel,
David Croydon and
Daisuke Shiraishi.
Preprint.

[ 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}},
}

*Chase-escape with death on trees*,
with
Erin Beckman,
Keisha Cook,
Nicole Eikmeier and
Matthew Junge.
Preprint.

[ 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}},
}

### Upcoming events

### Other writing

*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.