We thank PIMS for their support which makes this seminar series possible.
Speaker: Romain Panis (Lyon)
Time: 30 September, 2025 10:30-11:30 am (NOTE DIFFERENT TIME!)
Location: https://uvic.zoom.us/j/82117115633?pwd=T18dWUILT2zrhpDhsmMxJdbVkxLpW3.1
Title: A random walk approach to high-dimensional critical phenomena
Abstract: One of the main goals of statistical mechanics is to understand critical phenomena of lattice models. This can be achieved by computing the so-called critical exponents, which govern algebraic scaling near or at the critical point. This task is generally impossible due to the intricate interplay between the specific features of the models and the geometry of the graphs on which they are defined. A striking observation was made in the 20th century: above the upper critical dimension d_c, the geometry becomes inessential and critical exponents adopt their mean-field values (as on Cayley trees or complete graphs).
Classical approaches—renormalization group, differential inequalities with reflection positivity, and the lace expansion—are powerful yet model-specific and technically heavy. We revisit the study of the mean-field regime and introduce a unified, probabilistic framework that applies across perturbative settings, including weakly self-avoiding walk (d>4), spread-out Bernoulli percolation (d>6), and one- and two-component spin models (d>4).
Based on ongoing works with Hugo Duminil-Copin, Aman Markar, and Gordon Slade.
Speaker: Mathew Penrose (Bath, UK)
Time: 7 October, 2025 10:30 -11:30 am
Location: https://uvic.zoom.us/j/82117115633?pwd=T18dWUILT2zrhpDhsmMxJdbVkxLpW3.1
Title: Coverage and connectivity in stochastic geometry
Abstract: Consider a random uniform sample of size $n$ over a bounded
region $A$ in $R^d$, $d \geq 2$, having a smooth boundary.
The coverage threshold $T_n$ is the smallest $r$ such that the union $Z$
of Euclidean balls of radius $r$ centred on the sample points
covers $A$. The connectivity threshold $K_n$ is twice the smallest
$r$ required for $Z$ to be connected. These thresholds are random variables
determined by the sample, and are of interest, for example, in wireless
communications, set estimation, and topological data analysis.
We discuss recent results on the large-$n$ limiting distributions of $T_n$
and $K_n$. When $A$ has unit volume, with $v$ denoting the volume of the
unit ball in $R^d$ and $|dA|$ the perimiter of $A$, these take the form of
weak convergence of $ n v T_n^d - (2-2/d) \log n - a_d \log (\log n) $ to
a Gumbel-type random variable with cumulative distribution function
$$
F(x) = \exp (-b_d e^{-x} - c_d |dA| e^{-x/2}),
$$
for suitable constants $a_d, c_d$ with $b_2 =1$, $b_d =0 $ for $d>2$. The corresponding result for $K_n$ takes the same form with different constants $a_d, c_d$.
If time permits, we may also discuss extensions and related results
concerning (i) Other domains $A$ such as polytopes or manifolds;
(ii) coverage by balls of random radii;
(iii) strong laws
of large numbers for $T_n$ and $K_n$ for non-uniform random samples of points.
Some of the work mentioned here is joint work with Xiaochuan Yang and Frankie Higgs.
Speaker: Luca Michael Makowiec (U Leipzig)
Time: 14 October, 2025 10:30 am (NOTE DIFFERENT TIME!)
Location: https://uvic.zoom.us/j/82117115633?pwd=T18dWUILT2zrhpDhsmMxJdbVkxLpW3.1
Title: Random Spanning Trees in Random Environment
Abstract: We will introduce Random Spanning Trees in Random Environment (RSTRE), a disordered system on spanning trees that interpolates between the Uniform Spanning Tree (UST) and Minimum Spanning Tree (MST) measures. Our primary goal is to study how local and global observables of the RSTRE depend on the inverse temperature beta. Of particular interest is the diameter of a (typical) spanning tree, as it is the first step towards establishing convergence to a non-trivial scaling limit. Furthermore, for the RSTRE on the complete graph with uniform disorder, we discuss a sharp transition of the local limit of the RSTRE to either the UST or MST local limit. This stands in contrast to our conjectured smooth transition of the diameter when beta is inside the so-called intermediate regime.
Speaker:
Time: 21 October, 2025 2:30-3:20 pm
Location: TBA
Title:
Abstract
Speaker: Nathan Zelesko (Northeastern)
Time: 28 October, 2025 10:30-11:30 am (NOTE UNUSUAL TIME)
Location: https://uvic.zoom.us/j/82117115633?pwd=T18dWUILT2zrhpDhsmMxJdbVkxLpW3.1
Meeting ID: 821 1711 5633
Password: 469174
Title: Site percolation on planar graphs
Abstract: Site percolation models are probability distributions of 2-colorings of the vertices of locally finite infinite graphs. Historically, they have been studied on graphs exhibiting symmetries such as (quasi)-transitivity. In joint work with Alexander Glazman and Matan Harel, we instead only assume that the graph is planar. We show that a large class of site percolation models on any planar graph contains either zero or infinitely many infinite connected components. This includes the case of Bernoulli percolation at parameter p \leq 1/2, resolving a conjecture from the work of Benjamini and Schramm from 1996. In this talk, I will discuss the main ingredients of the proof, with emphasis on the properties we require of the model.
Speaker: Frederik R. Klausen (Princeton)
Time: 28 October, 2025 2:30-3:20 pm
Location: https://uvic.zoom.us/j/82117115633?pwd=T18dWUILT2zrhpDhsmMxJdbVkxLpW3.1
Meeting ID: 821 1711 5633
Password: 469174
Title: Title: Phase transitions of graphical representations of the Ising model
Abstract: Much of the recent rigorous progress on the classical Ising model was driven by new detailed understanding of its stochastic geometric representations.
Motivated by the problem of establishing exponential decay of truncated correlations of the supercritical Ising model in any dimension,Duminil-Copin posed the question in 2016 of determining the (percolative) phase transition of the single random current.
Using that the loop O(1) model is the uniform even graph of the random cluster model, we prove polynomial lower bounds for path probabilities (and infinite expectation of cluster sizes of 0) for both the single random current and loop O(1) model corresponding to any supercritical Ising model on the hypercubic lattice. The method partly extends to all positive integers q, where the analogue of the loop O(1) model is the q-flow model.
In this talk, I will introduce graphical representations of the Potts and Ising model and their many couplings followed by a discussion of new results whose surprising proof takes inspiration from the toric code in quantum theory.
Based on : https://link.springer.com/article/10.1007/s00220-025-05297-3 and https://arxiv.org/abs/2506.10765
Speaker: Ratul Biswas (NITMB)
Time: 4 November, 2025 3:30-4:20 pm (NOTE UNUSUAL TIME)
Location: Maclaurin D107 (NOTE UNUSUAL LOCATION)
Title: A unifying approach to the replica-symmetric regime in general dilute spin glasses
Abstract:
In this talk, I shall present a unifying approach to study the replica-symmetric regime of general dilute spin glasses. In particular, we identify two regimes - high temperature and subcritical dilution - where we observe a large class of popular spin glasses exhibiting replica-symmetric behavior. We revisit several well-known formulas for the free and ground state energies of some of these models and derive new ones for some others. Joint work with Arnab Sen and Wei-Kuo Chen.
Speaker: Lina Li (U Mississippi)
Time: 11 November, 2025 2:30-3:20 pm
Location: https://uvic.zoom.us/j/82117115633?pwd=T18dWUILT2zrhpDhsmMxJdbVkxLpW3.1
Meeting ID: 821 1711 5633
Password: 469174
Title: Lipschitz functions on weak expanders
Abstract: Given a connected finite graph $G$, an integer-valued function $f$ on $V(G)$ is called $M$-Lipschitz if the value of $f$ changes by at most $M$ along the edges of $G$. In 2013, Peled, Samotij, and Yehudayoff showed that random $M$-Lipschitz functions on graphs with sufficiently good expansion typically exhibit small fluctuations, giving sharp bounds on the typical range of such functions, assuming $M$ is not too large. We prove that the same conclusion holds under a relaxed expansion condition and for larger $M$, (partially) answering questions of Peled et al. Our approach combines Sapozhenko’s graph container method with entropy techniques from information theory.
This is joint work with Krueger and Park.
Speaker: Caelan Atamanchuk
Time: 18 November, 2025 2:30-3:20 pm
Location: https://uvic.zoom.us/j/82117115633?pwd=T18dWUILT2zrhpDhsmMxJdbVkxLpW3.1
Meeting ID: 821 1711 5633
Password: 469174
Title - The largest common subtree of two random trees
Abstract - Given two trees, what is the size and structure of their largest common shared subtree? This question has been a growing topic of interest in the probability/combinatorics community in recent years, with the problem having been discussed for a few different models of random trees. In this talk, we will discuss the case where the two trees are independent Bienaymé-Galton-Watson trees with finite-variance offspring distributions, conditioned to have size n. The main result will be a scaling limit for the size of the largest common subtrees of two such Bienaymé trees under some light assumptions. The talk is based on joint work with Omer Angel, Anna Brandenberger, Serte Donderwinkel, and Robin Khanfir.
Speaker: Hannah Cairns (McGill)
Time: 25 November, 2025 2:30-3:20 pm
Location: https://uvic.zoom.us/j/82117115633?pwd=T18dWUILT2zrhpDhsmMxJdbVkxLpW3.1
Title: Cooperative motion in higher dimensions
Abstract:
Cooperative motion is a random walk process where the jump rate of a particle depends on the likelihood that another independent identical walker is at the same position. It was studied on the line in a paper by Addario-Berry, Cairns, Devroye, Kerriou and Mitchell and two papers by Addario-Berry, Beckman, and Lin. We find the scaling limit for lattices $Z^d, d \ge 1$ in the special case of simple symmetric walk. This is the first result in dimension greater than one.
Speaker: Vasu Tewari (U Toronto (Mississauga))
Time: 1 December, 2025 3:30-4:20 pm (Part of the UVic undergrad colloquium)
Location: MACD 110
Title: Mixed Eulerian numbers: From probability to geometry
Abstract: Mixed Eulerian numbers were introduced by Postnikov in connection with the volumes of permutahedra. I will explain how these numbers appear in a simple probabilistic model involving particle-hopping, following work of Diaconis and Fulton. I will then explore how this perspective sheds light on geometric structures such as the permutahedral toric variety.
Speaker: Niki-Myrto Mavraki (University of Toronto (Mississauga))
Time: 2 December, 2025 3:30-4:20 pm
Location: DSB C118
Title:
Unlikely intersections from iterations: the riddle of shared preperiodic points
Abstract:
From Babylonian root-finding algorithms to Fatou-Julia's WWI-era fractals, iteration has always led to hidden patterns. But when do two different dynamical systems share many preperiodic points — numbers trapped in finite loops? We’ll explore this question rooted in the number theoretic theme of ‘unlikely intersections.’
Speaker:
Time: 6 january, 2026 2:30-3:20 pm
Location: TBA
Title:
Abstract
Speaker:
Time: 13 january, 2026 2:30-3:20 pm
Location: TBA
Title:
Abstract
Speaker: Ronnie Pavlov (Denver)
Time: 20 january, 2026 3:00-4:00 pm (NOTE UNUSUAL TIME!)
Location:
https://uvic.zoom.us/j/82117115633?pwd=T18dWUILT2zrhpDhsmMxJdbVkxLpW3.1
Meeting ID: 821 1711 5633
Password: 469174
Title: Maximal pattern complexity and pattern Sturmian sequences
Abstract: Maximal pattern complexity (MPC) of a subshift X was defined by Kamae and Zamboni in 2002 as a generalization of the usual word complexity function. It has nice connections to notions of independence in dynamics studied by Kerr and Li; for instance, subexponential growth of MPC implies that X is an almost 1-1 extension of a (compact abelian) group rotation.
At the opposite end of the spectrum, it is known that 2n is the minimum possible growth rate for MPC of an aperiodic sequence, and sequences achieving this are called pattern Sturmian. Simple Toeplitz sequences, codings of irrational rotations by 2 intervals, and some characteristic functions of sparse sets are known to be pattern Sturmian, and it has been an open question whether these are the only examples. I'll present some recent joint work with Anh Le and Casey Schlortt where we positively resolve this question, and describe some ideas of the proof and some open questions.
Speaker: Nathaniel Butler
Time: 27 january, 2026 2:30-3:20 pm
Location: COR B129
Title: Convergence in Distribution of Random Lipschitz Functions on d-ary Trees
Abstract: Take $n, M, d\geq 1$ as given. On the d-ary tree with n levels $T_d^n$, there are only finitely many functions $f:V(T_d^n)\rightarrow Z$ such that $|f(u)-f(v)|\leq M$ for any adjacent u and v and such that $f(v)=0$ for any leaf v. As such, one can take a uniform sampling of these functions and ask if anything interesting happens as $n\to\infty$. We have shown that the limit exists for certain small M and d, while the limit does not exist if d>>M log(M).
Speaker: Yinon Spinka (Tel Aviv)
Time: 27 january, 2026 3:30-4:20 pm
Location: COR B108 (NOTE UNUSUAL PLACE!!)
Title: Declocalization vs Localization of random Lipschitz functions on trees
Abstract:
A Lipschitz function on a graph G is a function f:V->Z from the vertex set V of the graph to the integers which changes by at most 1 along any edge of the graph. Given a finite connected graph G, and fixing the value of the function to be 0 on at least one vertex, we may sample such a Lipschitz function uniformly at random. What can we say about the typical height at a vertex? This depends heavily on G. For example, when G is a path of length n, and the height at one of the endpoints is fixed to be 0, this model corresponds to a simple random walk with uniform increments in {-1,0,1}, and hence the height at the opposite endpoint of the path is typically of order sqrt(n). In this talk, we consider the case when G is an infinite tree pruned at a given depth, and the height at the leaves is fixed to be 0. We would like to understand the distribution of the height at the root as the depth of the tree tends to infinity. A basic question is that of localization vs declocalization, that is, whether or not there is tightness of the height as the depth increases. Our main result is that the height function is localized if and only if the infinite tree is transient for simple random walk.
Joint work with Alon Heller.
Speaker: Dan Mikulincer
Time: 3 February, 2026 2:30-3:20 pm
Location: COR B129
Title:
Anti-concentration, from small-ball estimates to Fourier decay
Abstract: We will review different notions of what it means to be anti-concentrated, and look at how these definitions relate to each other, sometimes leading to surprising and nontrivial consequences. The talk will also cover applications ranging from algorithmic complexity to harmonic analysis and convex geometry.
Speaker: Arantha Ranu (UVic)
Time: 10 February, 2026 2:30-3:20 pm
Location: COR B 129
Title- Weakened Gibbs inequality for continuous potentials.
Abstract- It is well known that the Gibbs inequality, which says that the Gibbs ratio is bounded above and below by positive constants, holds for the unique equilibrium states of Hölder continuous potentials on shift spaces, but it can fail for continuous potentials. In this talk, we study the validity of a weaker form of the Gibbs inequality in this broader setting.
Speaker: Dylan Bansard-Tresse (UVic)
Time: 10 February, 2026 3:30-4:20 pm
Location: COR B112 (NOTE DIFFERENT PLACE)
Title: Returns to rare events for countable Markov shifts
Abstract: In this talk, we consider the question of quantitative recurrence in dynamical systems. Our aim is to describe how the law of return times to natural small targets evolves as they get smaller and to identify the corresponding limit behaviors. We will focus in particular on Countable Markov Shifts (CMS), a class of dynamical systems that generalizes both subshifts of finite type and Markov chains on countable state spaces. In the positive recurrent case, CMS exhibit a dichotomy, with the Poisson point process arising as the main limit. Our main interest, however, lies in the null recurrent case, where the Fractional Poisson process plays a central role. We will also show that, for natural targets, other limiting processes may appear.
Speaker: Shirshendu Ganguly (Berkeley)
Time: 12 February, 2026 3:30-4:20 pm (NOTE UNUSUAL DATE and TIME, part of Math undergraduate colloquium)
Location: COR A120
Title: Universality in nature
Abstract: A guiding principle in the study of various models arising in nature is that microscopic details of any particular model wash away on zooming out at the correct scale upon which a universal picture emerges dictated by certain broad mechanisms. This motivates classifying different models into universality classes. A well known such instance is the ubiquity of the Bell curve i.e. the Gaussian distribution in various statistical experiments or a Brownian motion as a universal random one dimensional curve.
In this talk we will explore universal features of a large class of models encompassing stochastic growth, forest fires, bacterial colonies, traffic flows, random geometry, directed polymers and random interfaces arising in ferromagnets. While predicted to exhibit characteristic non-Gaussian scaling behavior going back to the eighties, only very recently, a mathematically rigorous picture establishing some of the predictions has been formed relying on a variety of deep ideas, some of which will be reviewed.
The talk will be aimed at a general scientific audience and no particular background will be assumed.
Speaker:
Time: 17 February, 2026 2:30-3:20 pm (no seminar)
Location: TBA
Title:
Abstract
Speaker:
Time: 24 February, 2026 2:30-3:20 pm (no seminar)
Location: TBA
Title:
Abstract
Speaker: Peter Kosenko (UBC)
Time: 3 March, 2026 2:30-3:20 pm
Location: COR B129
Title: On recent progress related to studying harmonic measures for random walks on hyperbolic isometry groups and lamplighter groups
Abstract: Given a measured group $(G, \mu)$ acting on a nice space $X$, one might ask how to describe the
probability measures $\nu$ on $X$ which are invariant with respect to the action of $G$ on “average” with
respect to $\mu$. Turns out that such a measure ν usually is uniquely defined, depending on the choice
of $\mu$!
In my talk I will survey recent progress on understanding such measures with respect to actions of
isometry groups of hyperbolic spaces and lamplighter groups. In particular, we will discuss further
progress regarding obtaining the formulas for the Hausdorff dimensions of such measures. Joint
work with Eduardo Silva, Nikolay Bogachev and Giulio Tiozzo.
Speaker: Felix Clemen (UVic)
Time: 10 March, 2026 2:30-3:20 pm
Location: COR B129
Title: Regular Simplices in Higher Dimensions
Abstract:
A classical problem in combinatorial geometry, posed by Erdős in 1946, asks to determine the maximum number of unit segments in a set of n points in the plane. Since then a great variety of extremal problems in finite point sets have been studied. Here, we look at generalizations of this question concerning regular simplices. Among others we answer the following question asked by Erdős: Given n points in R^6, how many triangles can be equilateral triangles? For our proofs we use hypergraph Turán theory. This is joint work with Dumitrescu and Liu.
Speaker: Francesco Tosello (UBC)
Time: 17 March, 2026 2:30-3:20 pm
Location: COR B 129
Title: On the structure and stability of mixed memory attractors in the dynamics of the Hopfield model.
Abstract: The Hopfield model is a mean-field spin-glass model with structured random interactions. It was originally introduced as a model of associative memory because it can recover certain configurations - called pure memories - which are stored as global minima of its energy function and retrieved via a gradient-like dynamics. More generally, the attractors may be local minima corresponding to mixtures of several memories. However, the structure of these mixed attractors remains poorly understood, particularly from a rigorous mathematical perspective.
In this talk, I will present an ongoing work aimed at characterizing the critical points of the Hopfield energy landscape. This is done by analysing their stability with respect to the natural energy-decreasing dynamics of the model. In particular, I will establish a geometric condition ensuring dynamical stability and obtain partial results describing the behaviour near unstable mixed memories.
Speaker: Chris Hoffman (U Washington)
Time: 24 March, 2026 2:30-3:20 pm
Location: COR B129
Title: Activated Random Walk and Self-organized Criticality
Abstract: One of the most cited physics papers of the 1980s was Bak, Tang and
Wiesenfeld's introduction of self-organized criticality. This theory
said that diverse physical systems such as earthquakes, forest fires
and avalanches are all governed by similar underlying dynamics and
thus they share certain fundamental behaviors. They also proposed a
mathematical model that should have all of the properties that define
self-organized criticality. Such a model would be called a universal
model for self-organized criticality. Abelian networks are a class of
models that contain several good candidates for a universal model,
most notably activated random walk . We will discuss recent results
about activated random walk which give some progress to finding a
universal model for self-organized criticality.
This is joint work with Maddy Brown, Toby Johnson, Matt Junge, Joshua
Meisel and Hyojeong Son.
Speaker: Vilas Winstein (U Berkeley)
Time: 31 March, 2026 2:30-3:20 pm
Location: TBA
Title: Metastable wells in exponential random graph models
Abstract: Exponential random graph models (ERGMs) are exponential tilts of Erdos-Renyi models where higher-order interactions promote the presence of small subgraphs like triangles. These models exhibit metastable behavior at low temperatures (strong interaction), and decompose as mixtures of phase measures or metastable wells. We present a variety of novel results for these metastable wells, including quantitative CLTs and sharp bounds on the Wasserstein distance between an ERGM in a metastable well and a corresponding Erdos-Renyi model. A common thread in these results is the careful analysis of relevant quantities under a natural Markov chain sampling algorithm.