Welcome to my website!

My name is Avelio Sepúlveda and I am an assistant professor working at Universidad de Chile in the department of mathematics.

My main interests are probability theory and statistical physics. My research focuses on the study of the 2-dimensional Gaussian free field and its relationship with other concepts in random geometry and spin systems: SLE, CLE, Liouville measure, imaginary chaos, percolation and topological phase transitions. I am also interested in the study of random planar maps, in particular, in the model of tree-decorated planar maps and its scaling limits.

My e-mail is lsepulveda (at) dim (dot) uchile (dot) cl.

I am part of the ERC Vortex directed by Christophe Garban. I am an associate editor of Electronic Journal of Probability and Electronic Communications of Probability.

I co-organize the Seminario de Probabilidades de Chile and I am organizing the VII summer school on probabilities and Stochastic process in January 2025.

Background Image: Simulation of an ALE kindly done by Brent Werness.

An elementary approach to quantum length of SLE

with Ellen Powell

with Ellen Powell

In this work, we present an elementary proof establishing the equality of the right and left-sided $\sqrt{\kappa}$-quantum lengths for an SLE$_\kappa$ curve, where $\kappa \in (0,4]$. In words, take a free boundary GFF $\Phi$ in $\mathbb H$ and an independent SLE$_\kappa$ $\eta$ with $f_t:\mathbb H\backslash \eta([0,T])\to \mathbb H$ the associated conformal transformation with $f_T(\eta_T)=0$. The following equality is true: The $\sqrt{\kappa}/2$-chaos measure with respecto to the conformal change of coordinates of $\Phi$ of the set $f_T(\eta([s,t])\cap \mathbb R^+)$ is a.s. equal to $f_T(\eta([s,T]\cap \mathbb R^-)$ and furtheremore it is also equal to half the conformal Minkowski content of the curve, multiplied by $(4−\kappa)/2$ for $\kappa \in (0,4)$ and by 1 for $\kappa=4$. The first equality described for $\kappa\in (0,4)$ was first shown by Scott Sheffield in 2010, the second equality (without an identification of the constant) also for $\kappa \in (0,4)$ was shown in a work of Stephane Benoist in 2017. The proof of Sheffield is famously technical and abstract, and that of Benoist is strongly based in the proof of Sheffield. We provide an elementary proof of the result based in a new approximation of the conformal Minkowski content that better respects the conformal change of coordinates, we then have to show two things, that the left and right conformal Minkowski content is equal and that the limit of the approximations appropiately converges. The identification of the constant allows us to take $\gamma \nearrow 2$.

On the triviality of the shocked map

with Luis Fredes

with Luis Fredes

In this paper, we study the critical behaviour of the tree-decorated planar map. Sadly, we found that in this range of parameters the scaling limit is trivial, in the sense that when properly renormalised the metric space converges to a Brownian disk where the boundary is identified to a point. Furthermore, we show that for a tree-decorated planar quadrangulation where the map has $f$ faces and the tree has size $\sigma \sqrt{ f}$, the diameter of the tree is $\frac{f^{1/4}}{\log^\alpha(f)} \leq diam(Tree) \leq o(f^{1/4})$, where $\alpha$ is any number greater than $1$.

Excursion decomposition of the 2D continuum GFF

with Juhan Aru and Titus Lupu

with Juhan Aru and Titus Lupu

In this paper, we study the excursion decomposition of the GFF. That is to say, we write the GFF $\Phi$ as a sum of disjoint signed measures $\Phi=\sum \sigma_n \mu_n$, where $\sigma_n$ are i.i.d. Rademacher random variables. The construction of this decomposition is a consequence of the joint works with Juhan Aru and Titus Lupu. We also prove uniqueness of this decomposition under fairly weak assumptions and we show that it also appears as the scaling limit of the excursion decomposition of the GFF

The Gaussian free field (GFF) is the analogue of Brownian motion when the time set is changed by a 2-dimensional domain. Exit sets are the 2-dimensional analogue of exit times of the Brownian motion. Together with collaborators, we show existence and uniqueness of these sets. Furthermore, we study many of their properties and we relate them to the Brownian loop soup and CLE$_4$. The main difficulties of these works lie in the fact that the GFF is not a function, but only a Schwartz distribution.

On thin local sets of the Gaussian free field

Local sets are the d-dimensional counterpart of stopping times and have become a key to study properties of the Gaussian free field (GFF). In this work, we provide conditions on the size of a local set to show that it is thin, i.e. that the GFF does not put mass in it. Furthermore, we show that these conditions are tight (at least in dimension 2, 3 and 4) by constructing small local sets that are not thin.

On bounded-type thin local sets of the two-dimensional Gaussian free field

with Juhan Aru and Wendelin Werner

* in Journal of the Institute of Mathematics of Jussieu *

with Juhan Aru and Wendelin Werner

We introduce the first type of exit sets: two-valued thin local sets (TVS). We denote them $\mathbb A_{-a,b}$, and they are the 2-dimensional analogue of the first time a Brownian motion exits the interval $[-a,b]$. We show that they are unique and monotone and we give necessary and sufficient conditions for their existence ($a+b\geq 2\lambda = \pi$). We also show that any local set where the GFF remains bounded is a.s. contained in one of these sets. Furthemore, we prove that $\mathbb A_{-2\lambda,2\lambda}=\mathbb A_{-\pi,\pi}$ has the law of CLE$_4$, giving a new proof of the coupling between CLE$_4$ and the GFF proposed by Miller and Sheffield.

First passage sets of the 2D continuum Gaussian free field

with Juhan Aru and Titus Lupu

* in Probability Theory and Related Fields*

with Juhan Aru and Titus Lupu

We introduce the second type of exit sets: first passage sets (FPS). We call them $\mathbb A_{-a}$ and they are the 2-dimensional analogue of the first time a Brownian motion exits the interval $[-a,\infty)$. We show that they are unique and monotone and we show they exists for all $a>0$. Furthermore, we compute their size and we construct a measure that represent the values of the GFF in top of the FPS. In this paper, we also generalise exit sets for n-connected domains and general boundary conditions.

The first passage sets of the 2D Gaussian free field: convergence and isomorphisms

with Juhan Aru and Titus Lupu

* in Communications in Mathematical Physics *

with Juhan Aru and Titus Lupu

This paper is a continuation of the one above. We show the convergence of first passage sets (FPS) of the metric graph Gaussian free field to their continuum counterpart. We obtain numerous consequences: an easy proof of the convergence of a natural metric graph GFF interface to SLE$_4$, improved isomorphism theorems, computation of the size of a cluster of the critical loop soup and a new construction of the continuum FPS.

Two-valued local sets of the 2D continuum Gaussian free field: connectivity, labels, and induced metrics

with Juhan Aru

* in Electronic Journal of Probability *

with Juhan Aru

We study the connectivity properties of the loops of two-valued thin local sets (TVS) $\mathbb A_{-a,b}$, where a loop is defined as the boundary of a connected component of the complement of the set, i.e, $D\backslash \mathbb A_{-a,b}$. We show that these properties undergo a phase transition, from generating a connected graph when $a+b\in[-2\lambda,4\lambda)$, to generating a completely disconnected one when $a\geq 4\lambda$. In the first case, we study the graph metric and show that even though in the critical case the graph is not connected one can still make sense of a renormalised distance between these loops. As a consequence, we obtain a conformal invariant metric on the CLE$_4$ $\mathbb A_{-2\lambda,2\lambda}$. It is an open question to know whether this metric is measurable with respect to the CLE$_4$.

Dimension of two-valued sets via imaginary chaos

with Lukas Schoug and Fredrik Viklund

* in International Mathematics Research Notices *

with Lukas Schoug and Fredrik Viklund

We study the dimension of the two-valued thin local sets (TVS) and for each one of them we build a d-dimensional measure on top of it. The most interesting feature of this work is that the 2-point estimates are obtained by studying the behaviour of the imaginary multiplicative chaos close to a TVS. Furthermore, the $d$-dimensional measure is obtained as the restriction of the imaginary chaos, with the right parameter, to the TVS. In addition, this gives new proofs of the dimension of SLE$_4$ and CLE$_4$.

Extremal distance and conformal radius of a CLE$_4$ loop

with Juhan Aru and Titus Lupu

*in Annals of probability *

with Juhan Aru and Titus Lupu

In this paper, we use the theory of exit sets of the GFF to compute exact laws of geometric quantities related to the loop $\ell$ of a CLE$_4$ surrounding the origin. In particular, we compute the joint law between the conformal radius of the domain surrounded by $\ell$ together with the extremal distance from $\ell$ to the boundary. Futhermore, we obtain the same joint laws for the loop of all FPS' and all TVS' with $a+b=2k\lambda$. These laws are related with natural random times of Brownian motion. The main idea of this paper is that certain properties of the exit sets are better understood when the domain is not simply connected and that these properties may be understood in the simply connected domain by a coupling argument.

Percolation for 2D classical Heisenberg model and exit sets of vector valued GFF

with Juhan Aru and Christophe Garban

with Juhan Aru and Christophe Garban

In this paper, we study the exit sets of the discrete vector valued GFF in $d=2$. We show, by using techniques of continuous spin systems, that they are trivial in the sense that they are not of macroscopic size. Additionally, the study of the vectorial GFF allows us to revisit the objections of Patrascioiu and Seiler to the conjecture of Polyakov. We make part of their arguments rigorous and more importantly we provide the following counter-example using the vector value GFF: we build ergodic environments of (arbitrary) high conductances with (arbitrary) small and disconnected regions of low conductances in which, despite the predominance of high conductances, the XY model remains massive.

with Juhan Aru and Titus Lupu

In this paper, we study the excursion decomposition of the GFF. That is to say, we write the GFF $\Phi$ as a sum of disjoint signed measures $\Phi=\sum \sigma_n \mu_n$, where $\sigma_n$ are i.i.d. Rademacher random variables. The construction of this decomposition is a consequence of the joint works with Juhan Aru and Titus Lupu. We also prove uniqueness of this decomposition under fairly weak assumptions and we show that it also appears as the scaling limit of the excursion decomposition of the GFF

KT or topological phase transitions are a type of transition discovered by Kosterlitz and Thouless. Models that undergo this phenomenon are typically 2-dimensional and do not have a classical phase transition. We call them topological because at low temperatures the topological defects (vortices) of the model are local, however at large temperature these topological defects appear everywhere. The KT phase transition is in duality with the problem of localisation of integer-valued fields. This duality was famously used by Fröhlich and Spencer to formally prove the first case of topological phase transition in 1981. To do this, they show that the GFF conditioned to have integer values has fluctuations of order 1 at low temperatures, however it fluctuates as a constant times the GFF at large temperatures.

Quantitative bounds on vortex fluctuations in $2d$ Coulomb gas and maximum of the integer-valued Gaussian free field

with Christophe Garban

*in Proceedings of the London Mathematical Society *

with Christophe Garban

In this paper, we work with three models: the Villain model, the Coulomb gas and the integer-valued GFF. We study contribution to the fluctuation of the vortices of the models at low-temperature, for the Villain model and the Coulomb gas, and at high temperature for the integer-valued GFF. We show that the fluctuations coming from vortices are at least a constant times the contribution of the spin-wave. We show that the constant is at least of order $e^{-2\pi^2 \beta}$, and in fact we give good reasons for why it should be $e^{-\pi^2\beta}$.

Improved spin-wave estimate for Wilson loops in $U(1)$ lattice gauge theory

with Christophe Garban

*in International Mathematics Research Notices *

with Christophe Garban

In this paper, we use the techniques developed in the paper above to study the $U(1)$-lattice gauge theory in dimension $4$. This model is related to the study of the Yang-Mills theory for the commutative case and appears in the study of electromagnetisms. We show that in this case there is also a decoupling between the vortices and the spin-wave, and we prove that for any inverse-temperature the vortices fluctuates at least as a constant times the fluctuations of the spin-wave. We show that the constant is at least of order $e^{-2\pi^2 \beta}$, and in fact we give good reasons for why it should be $e^{-\pi^2\beta}$.

Statistical reconstruction of the Gaussian free field and KT transition

with Christophe Garban

*in Journal of the European Mathematical Society *

with Christophe Garban

In this work, we give a reinterpretation of the KT-transition. To do that we study the 2-dimensional discrete Gaussian free field, a model which originally does not have a (KT) phase transition. In this context, we reinterpret the KT-transition as an "informational" phase transition. To be more precise, imagine that you only have access to the fractionary part of the GFF and you want to recover its macroscopic observable up to an $o(1)$-error. We show that the KT-transition appears in the following way: if the temperature of this GFF is low enough, then this recovery is possible. On the other hand, when the temperature is high enough, there is no such a recovery process. This result is based in the study of the conditional fluctuations of the integer part of the GFF given its fractional part.

The theory of Gaussian multiplicative chaos (GMC) studies the exponential of the 2-dimensional continuum Gaussian free field (GFF). In fact, as the GFF is not a function the GMC cannot be defined pointwise, instead it is constructed as a measure by first discretising via mollification and then taking the limit. For a given level of mollification, the GMC measure has Radon-Nykodim derivative with respect to Lebesgue equal to the renormalised exponential of a GFF times a constant $\gamma$. GMC measures are fractal and they are non-trivial only when $\gamma\in (-2,2)$. Their importance comes from the fact that they are the conjectured scaling limit of the natural measure associated to planar maps weighted by statistical physics models.

Liouville measure as a multiplicative cascade via level sets of the Gaussian free field

with Juhan Aru and Ellen Powell

* in Annales de l'institut Fourier*

with Juhan Aru and Ellen Powell

We study the relationship between exit sets of the Gaussian free field (GFF) and Gaussian multiplicative chaos (GMC) measures. In particular, we show that GMC measures are multiplicative cascades where the branching mechanisms is random. This solves and generalises a conjecture due to E. Aïdekon. Furthermore, we show that with this way of writing we can also produce the critical measure, i.e., the case where the $\gamma=2$. In this case, a new normalisation is necessary. Let us also remark that this work has interesting consequences, as it has been used to unveil properties of the exit sets themselves.

Critical Liouville measure as a limit of subcritical measures

with Juhan Aru and Ellen Powell

* in Electronic Comunication of Probability *

with Juhan Aru and Ellen Powell

This work is a continuation of the one above. We use the fact that the Gaussian multiplicative chaos measures are multiplicative cascades to show that the left derivative at $\gamma=2$ is equal to minus two times the critical measure. This solves a conjecture by B. Duplantier, R. Rhode, S. Sheffield and V. Vargas.

with Ellen Powell

In this work, we present an elementary proof establishing the equality of the right and left-sided $\sqrt{\kappa}$-quantum lengths for an SLE$_\kappa$ curve, where $\kappa \in (0,4]$. In words, take a free boundary GFF $\Phi$ in $\mathbb H$ and an independent SLE$_\kappa$ $\eta$ with $f_t:\mathbb H\backslash \eta([0,T])\to \mathbb H$ the associated conformal transformation with $f_T(\eta_T)=0$. The following equality is true: The $\sqrt{\kappa}/2$-chaos measure with respecto to the conformal change of coordinates of $\Phi$ of the set $f_T(\eta([s,t])\cap \mathbb R^+)$ is a.s. equal to $f_T(\eta([s,T]\cap \mathbb R^-)$ and furtheremore it is also equal to half the conformal Minkowski content of the curve, multiplied by $(4−\kappa)/2$ for $\kappa \in (0,4)$ and by 1 for $\kappa=4$. The first equality described for $\kappa\in (0,4)$ was first shown by Scott Sheffield in 2010, the second equality (without an identification of the constant) also for $\kappa \in (0,4)$ was shown in a work of Stephane Benoist in 2017. The proof of Sheffield is famously technical and abstract, and that of Benoist is strongly based in the proof of Sheffield. We provide an elementary proof of the result based in a new approximation of the conformal Minkowski content that better respects the conformal change of coordinates, we then have to show two things, that the left and right conformal Minkowski content is equal and that the limit of the approximations appropiately converges. The identification of the constant allows us to take $\gamma \nearrow 2$.

Negative moments for Gaussian multiplicative chaos on fractal sets

with Christophe Garban , Nina Holden and Xin Sun

* in Electronic Comunication of Probability *

with Christophe Garban , Nina Holden and Xin Sun

We study the Gaussian multiplicative chaos (GMC) measures in the case where the base measure is not Lebesgue. We show that the total GMC mass has negative moments of all order. These result were already known in the case of Lebesgue measure, but the proof uses the scale invariance of the Gaussian free field (GFF) and the Lebesgue measure. We propose a different approach, were we do not use the scale invariance but the fact that one can describe the distribution of the GFF when one tilts its law by its total GMC mass. The main motivation was the study of the quantitative mixing properties of Liouville dynamical percolation (paper below).

The distance exponent for Liouville first passage percolation is positive

with Jian Ding and Ewain Gwynne

* in Probability Theory and Related Fields *

with Jian Ding and Ewain Gwynne

In this work, we study the distance exponent associated to the discrete Liouville first passage percolation (LFPP). That is to say take $\xi>0$ and a GFF $\phi$ in the discrete graph $[-2n,2n]^2$. Define the LFFP-distance as $\inf_{\gamma} \sum_{i \in \gamma} e^{\xi \phi(i)}$, where $\gamma$ is a path between $x$ and $y$. We show that if you take the distance between the inner and outer boundaries of an annulus of size $n$ is of order $n^\alpha$ for some $\alpha>0$. This is a crucial input in the proof that LFPP admits non-trivial subsequential scaling limits for all $\xi>0$.

Planar maps are a discrete approximation of a two-dimensional Riemmanian geometry. Mathematically, they are graph embedded on the sphere in such a way that edges do not cross. The metric structure of uniformly chosen planar maps have been deeply studied and their scaling limit correspond to a certain metric obtained from a GMC of parameter $\gamma=\sqrt{8/3}$. Recently, there has been a lot of interest in non-uniform laws of random planar maps. In particular, planar maps decorated by an FK-model. The simplest of this case correspond to the spanning-tree decorated planar map, which should converge, in an appropiate sense, to a GMC of parameter $\gamma=\sqrt 2$.

Tree-decorated planar maps

with Luis Fredes

* in The Electronic Journal of Combinatorics *

with Luis Fredes

Tree-decorated planar maps are a pair between a planar map and one given subtree of the map. They are a family that interpolates between planar maps, and spanning-tree decorated planar maps.
In this paper, we introduce tree-decorated planar maps and study their combinatorial properties. In particular, we relate them to maps with boundary, we obtain counting formulae for them and we show that the marginal law of tree in a uniformly chosen tree-decorated map is that of a uniformly chosen tree.

Scaling limit of random plane quadrangulation with a simple boundary, via restriction

with Jérémie Bettinelli , Nicolas Curien and Luis Fredes

* to appear in Annales de l'institut Henri Poincaré*

with Jérémie Bettinelli , Nicolas Curien and Luis Fredes

In this work, we introduce a new framework to show convergence of a family of conditional planar maps given that we know that the unconditionned family converges. To do this, we show that a uniformly chosen planar map with a simple boundary converges towards the Brownian disk. We do it by using that the uniformly chosen planar map with a (non-necessarily-simple) boundary also converges towards the Brownian disk and the fact that we have explicit counting formulas for the set of planar maps with simple boundary.

with Luis Fredes

In this paper, we study the critical behaviour of the tree-decorated planar map. Sadly, we found that in this range of parameters the scaling limit is trivial, in the sense that when properly renormalised the metric space converges to a Brownian disk where the boundary is identified to a point. Furthermore, we show that for a tree-decorated planar quadrangulation where the map has $f$ faces and the tree has size $\sigma \sqrt{ f}$, the diameter of the tree is $\frac{f^{1/4}}{\log^\alpha(f)} \leq diam(Tree) \leq o(f^{1/4})$, where $\alpha$ is any number greater than $1$.

The theory of percolation studies the connectivity properties of a random colouring of a graph. A natural dynamic on this model is to put independent Poissonian clocks in each vertex, and each time one of the rings, one resamples the colour of that vertex. In two dimension, and when the clocks all run at the same renormalised speed, this model has a non-trivial scaling limit.

Liouville dynamical percolation

with Christophe Garban , Nina Holden and Xin Sun

* in Probability Theory and Related Fields *

with Christophe Garban , Nina Holden and Xin Sun

In this work, we introduce the model of Liouville dynamical percolation. It is a dynamical percolation model in the triangular lattice in which the speed of each clock is given by a Liouville measure of parameter $\gamma$. We show that for all $\gamma\neq \pm\sqrt{3/2}$ this process has a non-trivial scaling limit which is mixing when $\gamma\in (-\sqrt{3/2},\sqrt{3/2})$ and frozen when $\gamma\notin[-\sqrt{3/2},\sqrt{3/2}]$.