# Probability Seminar

##### Organisers

Leonardo T. Rolla and Nikos Zygouras in collaboration with Sasha Sodin (Queen Mary), Ofer Busani and Benjamin Lees (Bristol).

##### Upcoming seminars

For the Zoom link, please contact the organisers the day before.

**July 07** 16:00 - Tyler Helmuth (Durham University)

**Title: The Arboreal Gas**

**Abstract:** In Bernoulli bond percolation each edge of a graph is declared open with probability p (and closed otherwise) and one typically asks questions about the random subgraph of open edges. The arboreal gas is the probability measure obtained by conditioning on the event that the subgraph of open edges is a forest, i.e., contains no cycles. What are the percolative properties of these random forests? Do they contain giant trees? This turns out to be a surprisingly rich question. I will discuss what is known and some conjectures.

##### Past seminars

**June 30** 16:00 - Sébastien Ott (University of Geneva)

**Title: **Failure of Ornstein-Zernike asymptotics for models with exponentially decaying interactions

**Abstract:** In 1914 and 1916, Ornstein and Zernike published a celebrated heuristic predicting the sharp asymptotic behaviour of density−density correlations in high temperature/low density gases. This theory has since then become a classical topic in statistical theory of fluids. The (non−rigorous) derivation of their asymptotics relies on a certain "mass gap condition", which was believed to hold at least in very high temperature/very low density regimes as soon as the potential is "short range" (exponentially decaying). In a recent joint work with Y. Aoun, D. Ioffe, and Y. Velenik, we proved that this condition can fail precisely in these regimes, when the interaction decays like ψ(r) e⁻ʳ. The failure/success of the condition (and of the OZ asymptotics) is then closely related to the pre−factor psi. In this talk, I will present the mechanism behind this failure, which can be seen as a condensation transition for a suitable model. To keep things elementary, I will restrict the discussion to the case of the Ising model.

**June 23** 16:00 - Laure Dumaz (ENS Paris)

**Title: **Localization of the continuous Anderson hamiltonian in 1-d and its transition towards delocalization

**Abstract:** We consider the 1−dimensional continuous Schrodinger operator − d²/dx² + B’(x) on an interval of size L where the potential B’ is a white noise. We study the entire spectrum of this operator in the large L limit. We prove the joint convergence of the eigenvalues and of the eigenvectors and describe the limiting shape of the eigenvectors for all energies. When the energy is much smaller than L, we find that we are in the localized phase and the eigenvalues are distributed as a Poisson point process. The transition towards delocalization holds for large eigenvalues of order L. In this regime, we show the convergence at the level of operators. The limiting operator is acting on R²−valued functions and is of the form "J ∂ₜ + 2*2 noise matrix'' (where J is the matrix ((0, −1)(1, 0))), a form which already appeared as a conjecture by Edelman Sutton (2006) for limiting random matrices. Joint works with Cyril Labbé.

**June 16** 16:00 - Hubert Lacoin (IMPA)

**Title:** Investigating the mixing time for the exclusion process in a random environment

**Abstract:** We consider a system of k particles on a segment of size N. Its rules of evolution are the following: a particle at site x∈ {1 ,..., N} jumps to x+1 with rate ωₓ and to x−1 with rate 1−ωₓ where ωₓ∈ (0,1), and a jump is cancelled if the site is already occupied. We consider the case where (ωₓ)ᴺₓ₌₁ is (the fixed realization of) a sequence of IID random variables. Assuming that 𝔼[log 1−ωₓ/ωₓ]≠ 0 (that is, transience of the random environment), we prove that this particle systems mixes fast, in the sense that the time that it requires for its distribution to get close to the equilibrium state grows like a power of N. We present a lower bound for the power in mixing time which we conjecture to be sharp. Joint work with S. Yang.

**June 09** 16:00 - Alexandre Stauffer (Roma Tre University)

**Title: **Non-equilibrium multi-scale analysis and coexistence in competing first-passage percolation

**Abstract:** We consider a natural random growth process with competition on ℤᵈ called first−passage percolation in a hostile environment, that consists of two first−passage percolation processes FPP₁ and FPP_λ that compete for the occupancy of sites. Initially FPP₁ occupies the origin and spreads through the edges of ℤᵈ at rate 1, while FPP_λ is initialised at sites called seeds that are distributed according to a product of Bernoulli measures of parameter p. A seed remains dormant until FPP₁ or FPP_λ attempts to occupy it, after which it spreads through the edges of ℤᵈ at rate λ. We will discuss the results known for this model and present a recent proof that the two types can coexist (concurrently produce an infinite cluster) on ℤᵈ. We remark that, though counterintuitive, the above model is not monotone in the sense that adding a seed of FPP_λ could favor FPP₁. A central contribution of our work is the development of a novel multi−scale analysis to analyze this model, which we call a multi−scale analysis with non−equilibrium feedback and which we believe could help analyze other models with non−equilibrium dynamics and lack of monotonicity. Based on a joint work with Tom Finn (Univ. of Bath).

**June 02** 16:00 - Slim Kammoun (Lancaster University) - CANCELLED

**May 26** 16:00 - Bruno Schapira (Aix-Marseille University)

**Title:** Large deviations for the intersection of two ranges

**Abstract:** It is well known that the ranges of two independent random walks on ℤᵈ have a finite intersection almost surely if and only if d is larger than or equal to 5. Now, what about the probability that the number of points visited by the two walks exceeds a large constant t? A famous result of Khanin, Mazel, Shlosman and Sinai from the early 90’s showed that it decays like a stretched exponential with the striking exponent 1−2/d, up to arbitrarily small error in the exponent. Our main result refines this asymptotic, and answers in the discrete setup a conjecture of van den Berg, Bolthausen, and den Hollander. We will explain this, and discuss the main ideas of the proof. This is joint work with Amine Asselah.

**May 19** 16:00 - Shirshendu Ganguly (University of California Berkeley)

**Title: **Fractal geometry in models of random growth

**Abstract:** In last passage percolation models predicted to lie in the Kardar-Parisi-Zhang (KPZ) universality class, geodesics are oriented paths moving through random noise accruing maximum weight. The weight of such geodesics as their endpoints are varied gives rise to an intricate random energy field expected to converge to a rich universal object known as the Directed Landscape constructed by Dauvergne, Ortmann and Virag. Reporting recent progress in our understanding of the random fractal geometry exhibited by the latter, we will discuss results about the coupling structure of the geodesic weight as the endpoints are varied. The talk will be based on joint works with subsets of R. Basu, E. Bates, A. Hammond and M. Hegde.

**May 12** 16:00 - Amanda Turner (Lancaster University)

**Title:** Growth of Stationary Hasting-Levitov

**Abstract:** Planar random growth processes occur widely in the physical world. One of the most well-known, yet notoriously difficult, examples is diffusion-limited aggregation (DLA) which models mineral deposition. This process is usually initiated from a cluster containing a single "seed" particle, which successive particles then attach themselves to. However, physicists have also studied DLA seeded on a line segment. One approach to mathematically modelling planar random growth seeded from a single particle is to take the seed particle to be the unit disk and to represent the randomly growing clusters as compositions of conformal mappings of the exterior unit disk. In 1998, Hastings and Levitov proposed a family of models using this approach, which includes a version of DLA. In this talk I will define a stationary version of the Hastings-Levitov model by composing conformal mappings in the upper half-plane. This is proposed as a candidate for off-lattice DLA seeded on the line. We analytically derive various properties of this model and show that they agree with numerical experiments for DLA in the physics literature. This talk is based on arXiv:2008.05792, which is joint work with Noam Berger and Eviatar Procaccia.

**May 05** 16:00 - Karen Habermann (University of Warwick)

**Title: **Fluctuations for Brownian bridge expansions and convergence rates of Lévy area approximations

**Abstract:** We start by deriving a polynomial expansion for Brownian motion expressed in terms of shifted Legendre polynomials by considering Brownian motion conditioned to have vanishing iterated time integrals of all orders. We further discuss the fluctuations for this expansion and show that they converge in finite dimensional distributions to a collection of independent zero-mean Gaussian random variables whose variances follow a scaled semicircle. We then link the asymptotic convergence rates of approximations for Brownian Lévy area which are based on the Fourier series expansion and the polynomial expansion of the Brownian bridge to these limit fluctuations. We close with the observation that the Lévy area approximation resulting from the polynomial expansion of the Brownian bridge is more accurate than the Kloeden-Platen-Wright approximation, whilst still only using independent normal random vectors.

**April 28** 16:00 - Sourav Chatterjee (Stanford University)

**Title:** New results for surface growth

**Abstract:** The growth of random surfaces has attracted a lot of attention in probability theory in the last ten years, especially in the context of the Kardar-Parisi-Zhang (KPZ) equation. Most of the available results are for exactly solvable one-dimensional models. In this talk I will present some recent results for models that are not exactly solvable. In particular, I will talk about the universality of deterministic KPZ growth in arbitrary dimensions, and if time permits, a necessary and sufficient condition for superconcentration in a class of growing random surfaces.

**April 21** 16:00 - Vadim Gorin (University of Wiscosin-Madison)

**Title:** Infinite beta random matrix theory

**Abstract:** Dyson's threefold approach suggests to deal with real/complex/quaternion random matrices as beta=1/2/4 instances of beta-ensembles. We complement this approach by the beta=∞ point, whose study reveals a number of previously unnoticed algebraic structures. Our central object is the G\inftyE ensemble, which is a counterpart of the classical Gaussian Orthogonal/Unitary/Symplectic ensembles. We encounter unusual orthogonal polynomials, random walks, and finite free polynomial convolutions.

**March 17** 16:00 - Vittoria Silvestri (University of Rome La Sapienza)

**Title:** How far do activated random walkers spread from a single source?

**Abstract:** Activated Random Walks (ARW) is an interacting particle system which is believed to be an example of Self-Organised Criticality (SOC), in that the competition between the particles' activity and inactivity is expected to spontaneously drive the system to a critical state. In this talk I will discuss several notions of criticality and, specialising to ARW on the one-dimensional lattice, I will illustrate how some of them relate to each other. Based on joint work with Lionel Levine (Cornell).

**March 10** 16:00 - Axel Saenz Rodriguez (University of Warwick)

**Title: **The ASEP and the Bethe Ansatz.

**Abstract:** The asymmetric simple exclusion process (ASEP) is an interacting particle system on a one dimensional lattice. It is a toy model for a driven diffusive system, and it belongs to the Kardar-Parisi-Zhang (KPZ) universality class. The Bethe Ansatz gives us an educated guess to write exact formulas for the probability function of the ASEP, leading to very precise results for certain of the limiting statistics of the ASEP. We will discuss some of these results for the ASEP on the line and on the ring.

**March 03** 16:00 - Jon Keating (University of Oxford)

**Title: **Moments of moments and Gaussian Multiplicative Chaos

**Abstract:** I will discuss connections between the moments of moments of the characteristic polynomials of random unitary (CUE) matrices and Gaussian Multiplicative Chaos, focusing in particular on the critical-subcritical case.

**February 24** 16:00 - Gady Kozma (Weizmann Institute of Science)

**Title:** The correlation length for percolation

**Abstract:** For near-critical percolation, there is a special scale, called the correlation length, such that below this scale the process looks like critical percolation, while above this scale the process looks like super- or subcritical percolation, as the case may be. We will discuss what is known about this value, with special emphasis on the most mysterious case, that of 3 dimensions. All terms would be explained in the talk.

**February 17** 16:00 - Davide Gabrielli (University of L'Aquila)

**Title:** Invariant measures of the Box Ball System having independent soliton components

**Abstract:** The Box-Ball System (BBS) is a one-dimensional cellular automaton on the integer lattice. It is related to the Korteweg-de Vries (KdV) equation and exhibits solitonic behaviour. It has been introduced by Takahashi and Satsuma, who identified conserved quantities called solitons. Ferrari, Nguyen, Rolla and Wang codify a configuration of balls by a double infinite array of integer numbers called the soliton components. We illustrate the original construction and an equivalent one based on a branching decomposition of the trees associated to the excursions of the walk of a ball configuration. Building over this codification, we give an explicit construction of a large family of invariant measures for the Box Ball System that are also shift invariant, including Markov and Bernoulli product measures. The construction is based on the concatenation of i.i.d. excursions of the associated walk trajectory. The corresponding random array of components has a product distribution with geometric marginals of parameters depending on the size of the solitons. The values of the parameters are obtained by a recursive equation. Joint work with P.A. Ferrari.

**February 10** 16:00 - Federico Camia (NYU-Abu Dhabi)

**Title: ** Scaling behavior and decay of correlations in the critical and near-critical planar Ising model

**Abstract:** The Ising model, introduced by Lenz in 1920 to describe ferromagnetism, is one of the most studied models of statistical mechanics. Its two dimensional version has played a special role in rigorous statistical mechanics since Peierls’ proof of a phase transition in 1936 and Onsager’s derivation of the free energy in 1944. This continues to be the case today, thanks to new results that make the connection between the planar Ising model and the corresponding Euclidean field theory increasingly more explicit. In this talk, I will introduce the Ising model and discuss recent results on its critical and near-critical scaling limits. I will focus in particular on the scaling behavior of the magnetization field at criticality and on the decay of correlations in the near-critical regime. (Based on joint work with R. Conijn, C. Garban, J. Jiang, D. Kiss, and C.M. Newman.)

**February 03** 16:00 - Jess Jay (University of Bristol)

**Title:** Interacting Particle Systems and Jacobi Style Identities

**Abstract:** The study of nearest neighbour interacting particle systems on the integer line has a rich history. Although most work in this area concerns the case of translation invariant measures, it can be fruitful to look beyond this case. In 2018, Balazs and Bowen considered product blocking measures for ASEP and ZRP. By relating the two they found a probabilistic proof of the Jacobi triple product identity, a well-known classical identity appearing throughout Mathematics and Physics. Naturally, one asks if other systems give rise to identities with combinatorial significance, via their blocking measures. In this talk we parameterise such a family and show that it gives rise to new 3 variable combinatorial identities. This is joint work with M. Balazs and D. Fretwell.

**January 27** 16:00 - Alexander Dunlap (Courant Institute of Mathematical Sciences)

**Title:** A forward-backward SDE from the 2D nonlinear stochastic heat equation

**Abstract:** I will discuss a two-dimensional stochastic heat equation in the weak noise regime with a nonlinear noise strength. I will explain how pointwise statistics of solutions to this equation, as the correlation length of the noise is taken to 0 but the noise is attenuated by a logarithmic factor, can be related to a forward-backward stochastic differential equation (FBSDE) depending on the nonlinearity. In the linear case, the FBSDE can be explicitly solved and we recover the log-normal behavior proved by Caravenna, Sun, and Zygouras. Joint work with Yu Gu (CMU).

**January 20** 16:00 - John Sylvester (University of Glasgow)

**Title:** Multiple Random Walks on Graphs: Mixing Few to Cover Many

**Abstract: **In this talk we will consider k random walks that are run independently and in parallel on a finite, undirected and connected graph. Alon et al. (2008) showed that the effect of increasing k, the number of walkers, does not necessarily effect the cover time (time until each vertex has been visited by at least one walk) in a straightforward manner. Despite subsequent progress in the area, the problem of finding a general characterisation of multiple cover times for worst-case start vertices remains an open problem.

We shall present some of our recent results concerning the multiple cover time from independent stationary starting vertices. Firstly, we improve and tighten various bounds, which allow us to establish the stationary cover times of multiple walks on several fundamental networks up to constant factors. Secondly, we present a framework characterising worst-case cover times in terms of stationary cover times and a novel, relaxed notion of mixing time for multiple walks called partial mixing time. Roughly speaking, the partial mixing time only requires a specific portion of all random walks to be mixed. Using these new concepts, we can establish (or recover) the worst-case cover times for many networks including expanders, preferential attachment graphs, grids, binary trees and hypercubes.

This is joint work with Nicolás Rivera and Thomas Sauerwald https://arxiv.org/abs/2011.07893

**January 13** 16:00 - Bálint Virág (University of Toronto)

**Title:** The heat and the landscape

**Abstract:** If lengths 1 and 2 are assigned randomly to each edge in the planar grid, what are the fluctuations of distances between far away points?This problem is open, yet we know, in great detail, what to expect. The directed landscape, a universal random plane geometry, provides the answer to such questions. In some models, such as directed polymers, the stochastic heat equation, or the KPZ equation, random plane geometry hides in the background. Principal component analysis, a fundamental statistical method, comes to the rescue: BBP statistics can be used to show that these models converge to the directed landscape.

**December 11:** Herbert Spohn (Munich)

**Title:** Generalized Gibbs measures of the Toda lattice

**Abstract:** According to statistical mechanics, Gibbs measures for a many-particle system are constructed from number, momentum, and energy, which are believed to be generically the only locally conserved fields. The classical Toda lattice is an integrable system and thus possesses a countable list of local conservation laws. Accordingly Gibbs measures of the Toda chain are indexed by an infinite set of parameters, rather than only three. This is the meaning of “generalised" in the title. Specifically for the Toda lattice, we will discuss the structure of generalised Gibbs measures and point out the connection to the one-dimensional log gas. This information is a central building block when writing down the appropriate Euler type equations for the Toda lattice.

**December 4:** Kieran Ryan (Queen Mary)

**Title:** The quantum Heisenberg XXZ model and the Brauer algebra

**Abstract:** Several quantum spin systems have probabilistic representations as interchange processes. Some of these processes can be thought of as continuous time random walks on the symmetric group, which in turn has led to study using representation theory. We will in particular discuss the spin-½ quantum Heisenberg XXZ model (and analogous models in higher spins, which are not the XXZ). This model, in a similar way to study using the symmetric group for other models, can be studied using the Brauer algebra. We will introduce this algebra, and its representation theory. Using this we obtain the free energy of the system when the underlying graph is the complete graph, which further lets us determine phase diagrams.

**November 27**: Horatio Boedihardjo (Warwick)

**Title:** The signature of a path

**Abstract: **There has been a lot of activities recently pingrov path-wise results for stochastic differential equations. Some of these activities has been inspired by rough path theory which enables integration against the irregular sample paths to be defined in a path-wise, as opposed to $L^2$, sense. While there have been many existence and uniqueness results, a natural next step is to develop descriptive path-wise theory for differential equations driven by specific path. In this talk, we study a particular example of differential equation and its solution, known as the signature. It is one of the simplest equations studied in rough path theory and has some interesting properties that general equations do not have.

**November 20:** Michael Damron (Georgia Tech)

**Title:** Critical first-passage percolation in two dimensions

**Abstract:** In 2d first-passage percolation (FPP), we place nonnegative i.i.d. weights $(t_e)$ on the edges of $Z^2$ and study the induced weighted graph pseudometric $T = T(x,y)$. If we denote by $p = P(t_e = 0)$, then there is a transition in the large-scale behavior of the model as $p$ varies from $0$ to $1$. When $p < 1/2, T(0,x)$ grows linearly in $x$, and when $p > 1/2$, it is stochastically bounded. The critical case, where $p = 1/2$, is more subtle, and the sublinear growth of $T(0,x)$ depends on the behavior of the distribution function of $t_e$ near zero. I will discuss my work over the past few years that (a) determines the exact rate of growth of $T(0,x)$, (b) determines the "time constant" for the site-FPP model on the triangular lattice and, more recently (c) studies the growth of $T(0,x)$ in a dynamical version of the model, where weights are resampled according to independent exponential clocks. These are joint works with J. Hanson, D. Harper, W.-K. Lam, P. Tang, and X. Wang.

**November 13:** Ewain Gwynne (Chicago)

**Title:** Tightness of supercritical Liouville first passage percolation

**Abstract:** Liouville first passage percolation (LFPP) with parameter $\xi >0$ is the family of random distance functions on the plane obtained by integrating $e^{\xi h_\epsilon}$ along paths, where $h_\epsilon$ for $\epsilon >0$ is a smooth mollification of the planar Gaussian free field. Previous work by Ding-Dub\'edat-Dunlap-Falconet and Gwynne-Miller showed that there is a critical value $\xi_{\mathrm{crit}} > 0$ such that for $\xi < \xi_{\mathrm{crit}}$, LFPP converges under appropriate re-scaling to a random metric on the plane which induces the same topology as the Euclidean metric: the so-called $\gamma$-Liouville quantum gravity metric for $\gamma = \gamma(\xi)\in (0,2)$. Recently, Jian Ding and I showed that LFPP admits non-trivial subsequential scaling limits for all $\xi > 0$. For $\xi >\xi_{\mathrm{crit}}$, the subsequential limiting metrics do \emph{not} induce the Euclidean topology. Rather, there is an uncountable, dense, Lebesgue measure-zero set of points $z\in\mathbb C $ such that $D_h(z,w) = \infty$ for every $w\in\mathbb C\setminus \{z\}$. We expect that the subsequential limiting metrics are related to Liouville quantum gravity with matter central charge in $(1,25)$. I will discuss the properties of the subsequential limiting metrics, their connection to Liouville quantum gravity, and several open problems.

**November 6:** Renan Gross (Weizmann)

**Title:** Stochastic processes for Boolean profit

**Abstract:** Not even influence inequalities for Boolean functions can escape the long arm of stochastic processes. I will present a (relatively) natural stochastic process which turns Boolean functions and their derivatives into jump-process martingales. There is much to profit from analyzing the individual paths of these processes: Using stopping times and level inequalities, we will reprove an inequality of Talagrand relating edge boundaries and the influences, and say something about functions which almost saturate the inequality. The technique (mostly) bypasses hypercontractivity. Work with Ronen Eldan. For a short, animated video about the technique (proving a different result, don't worry), see here: https://www.youtube.com/watch?v=vPLHAt_iv-0.

**October 30:** Perla Sousi (Cambridge)

**Title:** The uniform spanning tree in 4 dimensions

**Abstract:** A uniform spanning tree of ℤ⁴ can be thought of as the "uniform measure" on trees of ℤ⁴. The past of 0 in the uniform spanning tree is the finite component that is disconnected from infinity when 0 is deleted from the tree. We establish the logarithmic corrections to the probabilities that the past contains a path of length n, that it has volume at least n and that it reaches the boundary of the box of side length n around 0. Dimension 4 is the upper critical dimension for this model in the sense that in higher dimensions it exhibits "mean-field" critical behaviour. An important part of our proof is the study of the Newtonian capacity of a loop erased random walk in 4 dimensions. This is joint work with Tom Hutchcroft.

**October 23: **Inés Armendáriz (Buenos Aires)

**Title:** Gaussian random permutations and the boson point process

**Abstract:** We construct an infinite volume spatial random permutation associated to a Gaussian Hamiltonian, which is parametrized by the point density and the temperature. Spatial random permutations are naturally related to boson systems through a representation originally due to Feynman (1953). Bose-Einstein condensation occurs for dimensions 3 or larger, above a critical density, and is manifest in this representation by the presence of cycles of macroscopic length. For subcritical densities we define the spatial random permutation as a Poisson process of finite unrooted loops of a random walk with Gaussian increments that we call Gaussian loop soup, analogous to the Brownian loop soup of Lawler and Werner (2004). We also construct Gaussian random interlacements, a Poisson process of doubly-infinite trajectories of random walks with Gaussian increments analogous to the Brownian random interlacements of Sznitman (2010). For dimensions greater than or equal to 3 and supercritical densities, we define the spatial permutation as the superposition of independent realizations of the Gaussian loop soup at critical density and Gaussian random interlacements at the remaining density. We show some properties of these spatial permutations, in particular that the point marginal is the boson point process, for any point density.This is joint work with P.A. Ferrari and S. Yuhjtman.

**October 16:** Firas Rassoul-Agha (Utah)

**Title:** Geometry of geodesics through Busemann measures in directed last-passage percolation

**Abstract:** We consider planar directed last-passage percolation on the square lattice with general i.i.d. weights and describe geometry properties of the full set of semi-infinite geodesics in a typical realization of the random environment. The main tool is the Busemann functions viewed as a stochastic process indexed by the asymptotic direction. In the exactly solvable exponential model we give a complete characterization of the uniqueness and coalescence structure of the entire family of semi-infinite geodesics. Part of our results concerns the existence of exceptional (random) directions in which new interesting instability structures occur. This is joint work with Christopher Janjigian and Timo Seppalainen.

**October 8 (notice special day and time- Thursday, 2pm): **Naomi Feldheim (Bar Ilan)

**Title:** Persistence of Gaussian stationary processes

**Abstract: **Let *f* : ℝ→ℝ be a Gaussian stationary process, that is, a random function which is invariant to real shifts and whose marginals have multi-normal distribution. What is the probability that *f* remains above a certain fixed line for a long period of time? This simple question, which was posed by mathematicians and engineers more than 60 years ago (e.g. Rice, Slepian), has some surprising answers which were discovered only recently. I will describe how a spectral point of view leads to those results. Based on joint works with O. Feldheim, F. Nazarov, S. Nitzan, B. Jaye and S. Mukherjee.