David Aldous, Probability Seminar Sept 30, 2021 TITLE: Can one prove existence of an infectiousness threshold (for a pandemic) in very general models of disease spread? ABSTRACT: Intuitively, in any kind of disease transmission model with an infectiousness parameter, there should exist a critical value of the parameter separating a very likely from a very unlikely resulting pandemic. But even formulating a general conjecture is challenging. In the most simplistic model (SI) of transmission, one can prove this for an essentially arbitrary large weighted contact network. The proof for SI depends on a simple lemma concerning hitting times for increasing set-valued Markov processes. Can one extend to SIR or SIS models over similarly general networks, where the lemma is no longer applicable?

The inviscid Burgers' equation has the remarkable property that its dynamics preserve the class of spectrally negative L\'{e}vy initial data, as observed by Carraro and Duchon (statistical solutions) and Bertoin (entropy solutions). Further, the evolution of the L\'{e}vy measure admits a mean-field description, given by the Smoluchowski coagulation equation with additive kernel. In this talk we discuss ongoing efforts to generalize this result to scalar conservation laws, a special case where this is done, and a connection with integrable systems. Includes work with F. Rezakhanlou.

The last few years have witnessed an explosion in the number of mathematical models for random graphs and networks, as well as models for dynamics on these network models. In this context I would like to exhibit the power of two well known philosophies in attacking problems in random graphs and networks: First, local weak convergence: The idea of local neighborhoods of probabilistic discrete structures (such as random graphs) converging to the local neighborhood of limiting infinite objects has been known for a long time in the probability community and has proved to be remarkably effective in proving convergence results in many different situations. Here we shall give a wide range of examples of the above methodology. In particular, we shall show how the above methodology can be used to tackle problems of flows through random networks, where we have a random network with nodes communicating via least cost paths to other nodes. We shall show in some models on the completely connected network how the above methodology allows us to prove the convergence of the empirical distribution of edge flows, exhibiting how macroscopic order emerges from microscopic rules. Also, we shall show how for a wide variety of random trees (uniform random trees, preferential attachment trees arising from a wide variety of attachment schemes, models of trees from Statistical Physics etc) the above methodology shows the convergence of the spectral distribution of the adjacency matrix of theses trees to a limiting non random distribution function. Second, scaling limits: For the analysis of critical random graphs, one often finds that properly associated walks corresponding to the exploration of the graph encode a wide array of information (including the size of the maximal components). In this context we shall extend work of Aldous on Erdos-Renyi critical random graphs to the context of inhomogeneous random graph models. If time permits we shall describe the connection between these models and the multiplicative coalescent, arising from models of coagulation in the physical sciences.

Reliable information transmission between two sites of a network naturally leads to a percolation problem. When the information to be transmitted is quantum an exciting possibility arises: transform the network performing well chosen measurements to enhance the transmission probability. This idea, introduced recently by Acin, Cirac and Lewenstein is now systematically and successfully applied to a variety of two-dimensional networks, but open questions show that a complete theory is missing. The talk will involve some quanta, some network geometry, some percolation and, hopefully, some fun. No knowledge of quantum theory or percolation theory is assumed. Graduate students are encouraged to attend.

The talk will be an introduction to the world of functional inequalities with a geometric content. I will in particular focus on the family of log-Sobolev and Sobolev inequalities and show how these inequalities are closely connected to the geometry of the ambient space. I will mainly follow the Bakry-Ledoux approach to these inequalities which is is based on the notion of intrinsic curvature of a diffusion operator and at the end of the presentation will explain how these ideas have recently been used in sub-Riemannian geometry.

The partner model is an SIS epidemic in a population with random formation and dissolution of partnerships, and disease transmission only occurs within partnerships. Foxall, Edwards, and van den Driessche found the critical value and studied the subcritical and supercritical regimes. Recently Foxall has shown that (if there are enough initial infecteds) then the critical model survives for time \(O(N^{1/2})\). Here we improve that result by proving the convergence of \(i_N(t)=I(tN^{1/2})/N^{1/2}\) to a limiting diffusion. We do this by showing that in the first O(1), this four dimensional process collapses to two dimensions: the number of SI and II partnerships are constant multiples of the the number of infected singles \(I_t\). The other variable \(Y_t\), the total number of singles, behaves like an Ornstein-Uhlenbeck process on a time scale O(1) and averages out of the limit theorem for \(i_N(t)\). This is joint work with Anirban Basak and Eric Foxall.

9:00 Carl Mueller (Rochester) 10:30 David Herzog (Duke) 1:00 Elchanan Mossel (Berkeley)

A link to the paper can be found on her web page. In solid tumors, targeted treatments can lead to dramatic regressions, but responses are often short-lived because resistant cancer cells arise. The major strategy proposed for overcoming resistance is combination therapy. We present a mathematical model describing the evolutionary dynamics of lesions in response to treatment. We first studied 20 melanoma patients receiving vemurafenib. We then applied our model to an independent set of pancreatic, colorectal, and melanoma cancer patients with metastatic disease. We find that dual therapy results in long-term disease control for most patients, if there are no single mutations that cause cross-resistance to both drugs; in patients with large disease burden, triple therapy is needed. We also find that simultaneous therapy with two drugs is much more effective than sequential therapy. Our results provide realistic expectations for the efficacy of new drug combinations and inform the design of trials for new cancer therapeutics.

We propose a framework for modelling stochastic systems which satisfy detailed balance (or in other terminology, time-reversibility). Rather than specifying the dynamics through a state-dependent drift and diffusion coefficients, we specify an equilibrium probability density and a state-dependent diffusion coefficient. We argue that our framework is more natural from the modelling point of view and has a distinct advantage in situations where either the equilibrium probability density or the diffusion coefficient is discontinuous. We introduce a numerical method for simulating dynamics in our framework that samples from the equilibrium probability density exactly and elegantly handles discontinuities in the coefficients. This is joint work with Xin Yang.

We consider the ferromagnetic Ising model on a sequence of graphs $G_n$ converging locally weakly to a rooted random tree. Generalizing [Montanari, Mossel, and Sly (2012)], under an appropriate “continuity" property, we show that the Ising measures on these graphs converge locally weakly to a measure, which is obtained by first picking a random tree, and then the symmetric mixture of Ising measures with + and - boundary conditions on that tree. Under the extra assumptions that $G_n$ are edge-expanders, we show that the local weak limit of the Ising measures conditioned on positive magnetization, is the Ising measure with + boundary condition on the limiting tree. The “continuity" property holds except possibly for countably many choices of $\beta$, which for limiting trees of minimum degree at least three, are all within certain explicitly specified compact interval. We further show the edge-expander property for (most of) the configuration model graphs corresponding to limiting (multi-type) Galton Watson trees. This talk is based on a joint work with Amir Dembo.

We introduce a technique, based on perturbation theory for Hamiltonian PDEs, to derive the asymptotic equations of the motion of a free surface of a fluid over a rough bottom (one dimension). The rough bottom is described by a realization of a stationary mixing process which varies on short length scales. We show that the problem in this case does not fully homogenize, and random effects are as important as dispersive and nonlinear phenomena in the scaling regime. We will explain how these technique can be generalized to higher dimensions

Coupling is a way of constructing Markov processes with prescribed laws on the same space. The coupling is called Markovian if the coupled processes are co-adapted to the same filtration. We will first investigate Markovian couplings of elliptic diffusions and demonstrate how the rate of coupling (how fast you can make the coupled processes meet) is intimately connected to the geometry of the underlying space. Next, we will consider couplings of hypoelliptic diffusions (diffusions driven by vector fields whose Lie algebra span the whole tangent space). Constructing successful couplings (where the coupled processes meet almost surely) for these diffusions is a much more subtle question as these require simultaneous successful coupling of the driving Brownian motions as well as a collection of their path functionals. We will construct successful Markovian couplings for a large class of hypoelliptic diffusions. We will also investigate non-Markovian couplings for some hypoelliptic diffusions, namely the Kolmogorov diffusion and Brownian motion on the Heisenberg group, and demonstrate how these couplings yield sharp estimates for the total variation distance between the laws of the coupled diffusions when Markovian couplings fail. Furthermore, we will demonstrate how non-Markovian couplings can be used to furnish purely analytic gradient estimates of harmonic functions on the Heisenberg group by purely probabilistic means, providing yet another strong link between probability and geometric analysis. This talk is based on joint works with Wilfrid Kendall, Maria Gordina and Phanuel Mariano.

It was noticed experimentally in the late 90's that the speeds of traveling fronts in microscopic systems approximating the KPP equation converge unusually slowly to their continuum values. Brunet and Derrida made a very precise conjecture for the basic model equation, which is the KPP equation perturbed by white noise. We will explain the conjecture and sketch the main ideas of the proof. This is joint work with Carl Mueller and Leonid Mytnik.

The classical central limit theorem (CLT) states that for sums of a large number of i.i.d. random variables with finite variance, the distribution of the rescaled sum is approximately Gaussian. However, the statement of the central limit theorem doesn't give any quantitative error estimates for this approximation. Under slightly stronger moment assumptions, quantitative bounds for the CLT are given by the Berry-Esseen estimates. In this talk we will consider similar questions for CLTs for random walks in random environments (RWRE). That is, for certain models of RWRE it is known that the position of the random walk has a Gaussian limiting distribution, and we obtain quantitative error estimates on the rate of convergence to the Gaussian distribution for such RWRE. This talk is based on joint works with Sungwon Ahn and Xiaoqin Guo.

The frog model is a type of branching random walk model. Active "frogs" move according to random walks, and if they encounter a sleeping frog on their walk, the sleeping frog becomes active and begins an independent random walk. Over the past 20 years, recurrence properties and asymptotic behavior of this system (and many generalizations) have been studied extensively. One way to generalize this system is to consider the continuous version: Brownian motion frogs moving in R^d. In this talk, we will describe a continuous variant of the problem and show a limiting shape theorem analogous to prior discrete results.

In a world of polarized opinions on many cultural issues, we propose a model for the evolution of opinions on a large complex network. Our model is akin to the popular Friedkin-Johnsen model, with the added complexity of vertex-dependent media signals and confirmation bias, both of which help explain some of the most important factors leading to polarization. The analysis of the model is done on a directed random graph, capable of replicating highly inhomogeneous real-world networks with various degrees of assortativity and community structure. Our main results give the stationary distribution of opinions on the network, including explicitly computable formulas for the conditional means and variances for the various communities. Our results span the entire range of inhomogeneous random graphs, from the sparse regime, where the expected degrees are bounded, all the way to the dense regime, where a graph having n vertices has order n^2 edges.

Mean field game theory describes continuum limits of symmetric large-population games. These games can often be seen as competitive extensions of classical models of interacting particle systems, where the particles are now "controlled state process" (with application-specific interpretation, such as position, income, wealth, etc.). The coupled optimization problems faced by each process are typically resolved by Nash equilibrium, and there is a large and growing literature on solvability problems (both theoretical and computational). On the other hand, relatively little is known on how to rigorously pass from a finite population to a continuum, especially for dynamic stochastic games. The basic question is: Given for each N a Nash equilibrium for the N-player game, do the equilibria (more precisely, the empirical distributions of state processes) converge as N tends to infinity? This talk is an overview of the known probabilistic limit theorems in this context (law of large numbers, fluctuations, and large deviations), the ideas behind them, and some open problems.

In this talk I will review the basic ideas of the regularity structure theory developed by Martin Hairer, as well as its applications to stochastic PDE problems. I will then discuss my joint work with Hairer on the sine-Gordon equation and central limit theorems for stochastic PDEs.

The dynamics of a set of chemical reactions are usually modeled by mass action kinetics as a set of algebraic ordinary differential equations. This model sees the state space of the system as a continuum, whereas chemical reactions represent interactions of a discrete set of molecules. We study large fluctuations of the stochastic mass action kinetics model through Freidlin-Wentzell theory. The application of such a theory to this framework requires justification, in particular because of the non-uniformily Lipschitz character of the model. We therefore find, using tools of Lyapunov stability theory, a set of sufficient conditions for the applicability of large deviations theory to this framework, and prove that such conditions are satisfied by a large class of chemical reaction networks identified exclusively on the base of their topological structure.

In this talk we present two intimately related approaches in speeding-up molecular simulations via Monte Carlo simulations. First, we discuss coarse-graining algorithms for systems with complex, and often competing particle interactions, both in the equilibrium and non-equilibrium settings, which rely on multilevel sampling and communication. Second, we address mathematical, numerical and algorithmic issues arising in the parallelization of spatially distributed Kinetic Monte Carlo simulations, by developing a new hierarchical operator splitting of the underlying high-dimensional generator, as means of decomposing efficiently and systematically the computational load and communication between multiple processors. The common theme in both methods is the desire to identify and decompose the particle system in components that communicate minimally and thus local information can be either described by suitable coarse-variables (coarse-graining), or computed locally on a individual processors within a parallel architecture.

Consider a Galton-Watson tree. Pick two individuals at random by simple random sampling from the nth generation and trace heir lines of descent back in time till they meet. Call that generation X_n. In this talk we will discuss the probability distribution of X_n and its limits for the four cases m <1, m=1, m greater than 1 but finite, and m infinite, where m is the mean offspring size.

Consider a network of sites growing over time such that at step n a newcomer chooses a vertex from the existing vertices with probability proportional to a function of the degree of that vertex, i.e., the number of other vertices that this vertex is connected to. This is called a preferential attachment random graph. The objects of interest are the growth rates for the growth of the degree for each vertex with n and the behavior of the empirical distribution of the degrees. In this talk we will consider three cases: the weight function w(.) is superlinear, linear, and sublinear. Using recently obtained limit theorems for the growth rates of a pure birth continuous time Markov chains and an embedding of the discrete time graph sequence in a sequence of continuous time pure birth Markov chains, we establish a number of results for all the three cases. We show that the much discussed power law growth of the degrees and the power law decay of the limiting degree distribution hold only in the linear case, i.e., when w(.) is linear

It is known that the simple random walk on the unique infinite cluster of supercritical percolation on Z^d diffuses in the same way it does on the original lattice. In critical percolation, however, the behavior of the random walk changes drastically. The infinite incipient cluster (IIC) of percolation on Z^d can be thought of as the critical percolation cluster conditioned on being infinite. Alexander and Orbach (1982) conjectured that the spectral dimension of the IIC is 4/3. This means that the probability of an n-step random walk to return to its starting point scales like n^{-2/3} (in particular, the walk is recurrent). In this work we prove this conjecture when d>18; that is, where the lace-expansion estimates hold. Joint work with Gady Kozma.