Research Seminar

 Abstract: We establish local boundedness, parabolic weak Harnack inequalities and H\"{o}lder regularity of weak solutions to the parabolic equation associated with the infinitesimal generator of Markov chains on $\mathds{Z}^d$ (with $d\geq3$) with long-range jumps which have finite second-order moments. The proof is based on De Giorgi's iteration and Moser's iteration.

 Abstract: The mathematical problem of homogenization typically involves studying the effective parameters of a system that exhibits rapid variations in its spatial characteristics. However, we focus on a stochastic multivariate homogenization problem of a different kind: the diffusion in the presence of narrowly located semipermeable interfaces. In simple words, our model reminds of a foiled composite material consisting of a media interlaced with very thin plates of different permeability. In material science such models are referred to as reinforced materials like a glass wool reinforced by aluminium foil. Usually, one is interested in the effective parameters of such a system. By combining the study of stochastic differential equations with local times and homogenization, we explore how the presence of interfaces can alter the diffusion behavior of the limit process. 

 As a byproduct of our research, we obtain theorems for the existence and uniqueness of solutions to SDEs for multidimensional diffusion processes with membranes. Uniqueness is a problem of particular interest because it implies the strong Markov property of the solution, which is essential for the proof of convergence.

 Abstract: In 1972, Lieb and Robinson showed that quantum information propagates ballistically in systems with local and bounded interactions, a result now known as the Lieb-Robinson bounds (LRBs). Since the early 2000s, Hastings and others have used LRBs as a powerful tool to resolve long-standing open problems in mathematical physics, prompting much work on generalizing them. However, the understanding of LRBs for unbounded interactions remains incomplete. A key example is the Bose-Hubbard model, where standard techniques fail due to boson accumulation. Schuch, Harrison, Osborne, and Eisert suggested that controlling boson transport may be central to extending LRBs in this context. It is then crucial to control higher moments of the number operator, as that allows us to employ Markov inequality to control fluctuations. Following this idea, many results have been proved, but long-range unbounded interactions still constitute a big obstacle.

In this talk, I will present a new particle propagation bound that could allow improving LRB for long-range interactions. The main novelty in our proof is a multi-scale adaptation of the adiabatic space-time localisation observable (ASTLO) method, which allows removing the dependence of the error term from far-away particles.

 Abstract: In this talk, we discuss the derivation of nonlinear effective equations, such as the nonlinear Schrödinger and Hartree equations, from linear many-body Schrödinger dynamics. After introducing the physical motivation and mathematical framework, we will explore the key concepts and tools that enable a rigorous approximation of complex many-body quantum evolution by simpler nonlinear PDEs. In particular, we will highlight the role of dispersive and Strichartz-type estimates in controlling the approximation and understanding the emergent dynamics.

 Abstract: Voiculescu's notion of asymptotic free independence applies to a wide range of random matrices, including those that are independent and unitarily invariant. In this talk, we generalize this notion by considering random matrices with a tensor product structure that are invariant under the action of local unitary matrices. Assuming the existence of the “tensor distribution” limit described by tuples of permutations, we show that an independent family of local unitary invariant random matrices satisfies asymptotically a novel form of independence, which we term “tensor freeness”. Furthermore, we propose a tensor free version of the central limit theorem, which extends and recovers several previous results for tensor products of free variables. This is joint work with Ion Nechita.

 Abstract: We consider a one-dimensional microscopic reaction-diffusion process obtained as a superposition of a Glauber and a Kawasaki dynamics. The reaction term is tuned so that a dynamical phase transition occurs in the model as a suitable parameter is varied. We study dynamical fluctuations of the particle's density at the critical point. We characterise the slowdown of the dynamics at criticality, prove that density fluctuations are non-Gaussian and characterise their limit as the solution of a non-linear SDE. The proof relies on a decoupling of slow and fast scales relying in particular on a relative entropy argument. Joint work with Benoit Dagallier.

 Abstract: Last passage percolation is a representative model of phenomena that characterize the Kardar-Parisi-Zhang universality class. The model has an interesting integrable structure that allows for the computation of many surprising exact formulas, based on connections with random matrices, representation theory, orthogonal polynomials, and more. The last 5-6 years has even seen the construction of LPP's universal scaling limit, called the directed landscape.  

 This talk focuses on the pre-limiting lattice model, specifically on large deviations for its geodesic paths. As for simple random walk, the large deviations formulas are not universal. Instead they depend on the distribution of the iid inputs, which is an aspect of the general problem that is much less studied. I will describe our characterization of these large deviations formulas in terms of last passage times. For integrable choices of the iid weights these characterizations lead to very explicit formulas, which allow us to directly compare the large deviations for the geodesic to those of random walk paths.

 Based on joint work with Riddhi Basu, Sean Groathouse, and Xiao Shen.

 Abstract: One of the fascinating phenomena in spin glasses is the dramatic change in behavior between the high- and low-temperature regimes. In this talk, examine a variant of the famous Sherrington-Kirkpatrick (SK) model, known as the bipartite spherical SK model. We focus on the critical temperature threshold and present results on the fluctuations of the free energy. We will highlight the key strategies and tools from random matrix theory used in analyzing the model. Time permitting, we will compare this model to the bipartite model with discrete spins, where much less is known. The talk is based on joint work with E. Collins-Woodfin.

 Abstract: We prove the strong convergence of SDEs approximating by compound Poisson processes, and give the rate of convergence. For SDEs with diffusion coefficient $\alpha$-H\"older continuous on the time variable, the convergence has polynomial rate. For singular stochastic Volterra equations, the convergence has logarithmic rate. Compared with Euler approximation, we do not need H\"older continuity assumption on the time variable of the drift coefficient for compound Poisson approximation.

 Abstract: In the large N limit, complex and symplectic models exhibit limiting spectra whose support is called the droplets. In this talk, we will discuss the elliptic Ginibre matrices conditioned to have real eigenvalue with multiplicity proportional to the dimension of the matrix. We prove that the droplets are either simply connected, doubly connected, or composed of two simply connected components. Moreover, we present the explicit description of the droplet and electrostatic energies for the simply and doubly connected case. Finally, we introduce the asymptotic behavior of the moments of the characteristic polynomials of elliptic Ginibre matrices as an application. This is based on a joint work with Sung-Soo Byun. (arXiv:2502.02948)

 Abstract: A geodesic triangle $\Delta(x,y,z)$ in a metric space is called $\delta$-slim if every point on any side of the triangle lies within $\delta$ distance of one of the other two sides. Equivalently, the slimness of the triangle is defined as the infimum of such $\delta$. In this talk, we explore the slimness of geodesic triangles in random graphs, analyzing the expected value of slimness as a measure of how "slim" or "fat" these triangles tend to be. We will also discuss how this analysis provides insights into the geometric and combinatorial properties of random graphs, with connections to broader aspects of metric geometry. Based on joint work with Anna Gilbert.

 Abstract: In this talk, we study regular symmetric Dirichlet forms without killing on general metric measure spaces with doubling measures. For local Dirichlet forms, it is known that the on-diagonal upper heat kernel estimate (DUE) is equivalent to a weak localized Faber-Krahn inequality (FK). The Faber-Krahn inequality is also known to be equivalent to a local Nash inequality. When the index of the scale function exceeds 2, a cut-off Sobolev inequality is additionally required. For non-local Dirichlet forms, a similar result has been established under the assumption of a pointwise upper bound on the jump density.

We extend this analysis to the case where the jump kernel is not absolutely continuous. In this context, we show that the equivalence between DUE and FK fails. We then introduce a condition under which this equivalence holds, provided the cut-off Sobolev inequality is satisfied. Finally, we prove that this condition cannot be improved without further assumptions.

 Abstract: Liouville quantum gravity (LQG) is a "canonical" one-parameter model of surfaces with random geometry, where the parameter c >1 is the central charge of the associated conformal field theory. Compared to the subcritical and critical phases with c ≥ 25 (corresponding to 𝛾 ≤ 2), much less is known about the geometry of LQG in the supercritical phase c ∈ (1,25). Recent work of Ding and Gwynne has shown how to construct LQG in this phase as a planar random geometry associated with the Gaussian free field, which exhibits "infinite spikes." In contrast, a number of results from physics, dating back to the 1980s, suggest that supercritical LQG surfaces should look like the continuum random tree. 

In this talk, I will give a result that reconciles these two descriptions. More precisely, for a family of random planar maps in the universality class of supercritical LQG, if we condition on the (small probability) event that the planar map is finite, then the scaling limit is the continuum random tree. Separately, we show that there does not exist any locally finite measure associated with supercritical LQG which is locally determined by the field and satisfies the LQG coordinate change formula. Both results are based on a branching process description of supercritical LQG which comes from its coupling with CLE_4 by Ang and Gwynne. This is joint work with Manan Bhatia and Ewain Gwynne.

 Abstract: In this talk, we introduce the concept of thresholds for random graphs. After examining a simple example, we discuss the most fundamental graph properties that exhibit a sharp threshold: the emergence of a giant component and connectivity. We shall consider an inhomogeneous random graph and explain the main ideas used in the proof for the threshold of connectivity. 

 Abstract: The study of random graphs was initiated by Erdös and Rényi in the late 1950s. The classical Erdös-Rényi random graph is notable for its intriguing and elegant properties. However, in this model, all the vertices are equivalent in some sense, making it inadequate for modeling real-world networks, which are highly heterogeneous. To address this limitation, various inhomogeneous random graphs have been constructed and studied. In this talk, we begin by exploring the Erdös-Rényi random graph and its key properties, then examine an inhomogeneous model. We discuss similar properties between these two models and present a new method used to prove the phase transitions concerning the size of the largest component and connectivity in the inhomogeneous model.

 Abstract: The martingale problem is an efficient way to characterise a Markov process by its generator without having to take the diversions via the semigroup. In the first talk, we describe Strook and Varadhan's idea of using the martingale property as a useful tool to construct Markov processes and the advantages that this approach offers. In the second talk, we will apply the technique more specifically to the situation of non-local generators. In particular, the representation of the generators as pseudo-differential operators will play a role.

 Abstract: The martingale problem is an efficient way to characterise a Markov process by its generator without having to take the diversions via the semigroup. In the first talk, we describe Strook and Varadhan's idea of using the martingale property as a useful tool to construct Markov processes and the advantages that this approach offers. In the second talk, we will apply the technique more specifically to the situation of non-local generators. In particular, the representation of the generators as pseudo-differential operators will play a role.

 Abstract: In this presentation, we will investigate the support properties of solutions to nonlinear stochastic heat equations with random noise. The noise may be either space-time white noise or spatially homogeneous colored noise satisfying the reinforced Dalang’s condition. We will introduce conditions on the diffusion coefficient that guarantee the compact support property and strictly positive of solutions. Moreover, we will propose potential extensions of these conditions. This work is a collaboration with Kunwoo Kim and Jaeyun Yi.

 Abstract: I will discuss the asymptotic expansions of the Euclidean Φ^4-measure in the low-temperature regime. Consequently, we derive limit theorems, specifically the law of large numbers and the central limit theorem for the Φ^4-measure in the low-temperature limit. In the second part of the talk, I will focus on the infinite volume limit of the focusing Φ^4-measure. Specifically, with appropriate scaling, the focusing Φ^4-measure exhibits Gaussian fluctuations around a scaled solitary wave, that is, the central limit theorem.

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 Abstract: In this talk, we discuss the relationship between the precise asymptotics of the level-two large deviation rate function and the behavior of metastable stochastic systems. It is conjectured that the metastable and pre-metastable behaviors are encapsulated in the level-two large deviation of the markov process of the stochastic system. We rigorously prove this statement for a certain class of zero-range process by developing a methodology for $\Gamma$-convergence in the pre-metastable time scale and linking the resolvent approach to metastability (Landim et al., J Eur Math Soc, 2023. arXiv:2102.00998) with the $\Gamma$-expansion in the metastable time scale.

 Abstract: In this talk, we present a precise estimate for the mean extinction time of the contact process with a fixed infection rate on a star graph with $N$ leaves. Specifically, we determine not only the exponential main factor but also the sharp sub-exponential prefactor of the asymptotic formula for the mean extinction time as $N\to\infty$, a level of detail previously available only for complete graphs. We achieved these results by estimating the quasi-stationary distribution on non-extinction using special function theory and refined Laplace's method, and by applying the potential theoretic approach to metastability of non-reversible Markov processes. This novel integration of methodologies provides new insights into the study of the contact process.

 Abstract: The calculation of the asymptotics of the probability that there are no particles in a certain gap, known as the gap probability, is an important problem in point processes. In this talk, I will present the asymptotic expansion of the gap probabilities of complex and symplectic induced spherical ensembles, which can be realized as determinantal and Pfaffian 2D Coulomb gases on the Riemann sphere with the insertion of point charges. More precisely, when the gap is a spherical cap around the poles, we show that the gap probability has an asymptotic behavior of the form \[ \exp(C_1n^2 + C_2n\log n + C_3n + C_4\sqrt{n} + C_5\log n+ C_6 + o(1)). \] Our proof relies on the uniform asymptotics of the incomplete beta function. This is based on joint work with Sung-Soo Byun.

 Abstract: Contact process is a class of interacting particle systems which models the spread of an infection in a population. In this talk, we explore former results on the metastable behavior of contact processes, and look into the Eyring–Kramers law for contact process on stars.

 Abstract: The extremal process of branching Brownian motion (BBM)--- i.e., the collection of particles furthest from the origin-- has gained lots of attention in dimension $d = 1$ due to its significance to the universality class of log-correlated fields, as well as to certain PDEs. In recent years, a description of the extrema of BBM in $d > 1$ has been obtained. In this talk, we address the following geometrical question that can only be asked in $d > 1$. Generate a BBM at a large time, and draw the outer envelope of the cloud of particles: what is its shape? Macroscopically, the shape is known to be a sphere; however, we focus on the outer envelope around an extremal point-- the "front" of the BBM. We describe the scaling limit for the front, with scaling exponent 3/2, as an explicit, rotationally-symmetric random surface. Based on joint works with Julien Berestycki, Bastien Mallein, Eyal Lubetzky, and Ofer Zeitouni.

 Abstract: McKean-Vlasov SDEs is also called distribution-dependent SDEs, or mean-field SDEs. It's the macrocosmic limit of interacting particle systems and the probabilistic interpretation of non-linear Fokker-Planck PDEs.  

 Abstract: In this presentation, I'll explore the analysis of non-local regular Dirichlet forms on metric measure spaces. Our approach relies on three key assumptions: the presence of a strongly local Dirichlet form with sub-Gaussian heat kernel estimates, a tail estimate governing the jump measure outside balls, and a local energy comparability condition. Our primary objectives include establishing function inequalities such as localized Poincaré, cutoff Sobolev, and Faber-Krahn inequalities, highlighting their inherent stability structures and their implications for the regularity of corresponding harmonic functions. Our results are robust in the sense that the constants in estimates remain bounded, provided that the order of the scale function appearing in the tail estimate and local energy comparability condition, maintains a certain distance from zero.

 Abstract: In this talk, we provide sensitivity analysis for two types of problems: robust optimization and nonlinear Kolmogorov partial differential equations (PDEs). Our first goal is to quantify the sensitivity of a given robust optimization problem to model uncertainty. This can be achieved by showing that the robust problem can be approximated as ε approaches 0 by the baseline problem, computed using baseline processes. Subsequently, we aim to quantify the sensitivity of these PDEs to small nonlinearities in the Hamiltonian of drift and diffusion coefficients, and then use the results to develop an efficient numerical method for their approximation. We demonstrate that as ε approaches 0, the nonlinear Kolmogorov PDE can be approximated by linear Kolmogorov PDEs involving the baseline coefficients. Our derived sensitivity analysis then provides a Monte Carlo-based numerical method that can efficiently solve these nonlinear Kolmogorov PDEs. This talk is based on joint works with Daniel Bartl (Univ. Vienna) and Ariel Neufeld (NTU Singapore).

 Abstract: In this talk we present a quantum Monte Carlo algorithm to solve high-dimensional Black-Scholes PDEs with correlation for high-dimensional option pricing. The payoff function of the option is of general form and is only required to be continuous and piece-wise affine (CPWA), which covers most of the relevant payoff functions used in finance. We provide a rigorous error analysis and complexity analysis of our algorithm. In particular, we prove that the computational complexity of our algorithm is bounded polynomially in the space dimension d of the PDE and the reciprocal of the prescribed accuracy ε. Moreover, we show that for payoff functions which are bounded, our algorithm indeed has a speed-up compared to classical Monte Carlo methods. Furthermore, we present numerical simulations in one and two dimensions using our developed package within the Qiskit framework tailored to price CPWA options with respect to the Black-Scholes model, as well as discuss the potential extension of the numerical simulations to arbitrary space dimension. This talk is based on joint work with Jianjun Chen and Yongming Li.

 Abstract: This presentation explores the link between metastability and large deviation rate functions, focusing on level two LDP's role in defining metastable and premetastable timescales. It will also introduce applications of these concepts to the zero-range process.

 Abstract: We introduce the concept of a hyperbolic Fourier series and apply the main representation theorem to the solutions of the Klein-Gordon equation in 1+1 dimensions.

 Abstract: Estimation of exit time of an additive Markov process which is closely related to the convergence speed of Stochastic Gradient Descent.

 Abstract: In this talk, we will analyze a small random perturbation of dynamical system given by ordinary differential equation which is described by a stochastic differential equation $dx_{t}^{\epsilon}=-\nabla U(x_{t}^{\epsilon})dt+\epsilon dL_{t}$, where $L_{t}$ is a heavy-tailed Levy process. For this model, we provide the sharp asymptotics on the exit time from the interval containing the local minimum of $U$.

 Abstract: Metastability, characterized by transitions between (meta)stable states, is a widespread phenomenon observed in various stochastic systems within the low-temperature regime. In this presentation, we introduce the resolvent approach as a key method for understanding the metastability and its application to Langevin dynamics.

 Abstract: We study solutions of the irregular Stratonovich SDE $dX =|X|^\alpha \circ dB$, $\alpha\in (0, 1)$. In particular, we construct solutions spending positive time in 0, describe solutions spending zero time in 0, and show how a particular physically natural solution can be singled out by means of an additional external "ambient" noise.

This talk is based on the joint works with G. Shevchenko (Kiev).

 Abstract: The spectral heat content (SHC) measures the total heat that remains on a domain when the initial temperature is one and the outside temperature is identically zero. When one replaces the Laplace operator in the heat equation with generators of Lévy processes, one obtains SHC for those Lévy processes. Recently, the two-term asymptotic behavior of SHC for isotropic stable processes on bounded C^11 open sets was investigated by Park and Song (EJP 2022). In this talk, we generalize their result to cover Lévy processes with regularly varying characteristic exponent with index in (1,2]. The proof provides a unified approach to the study of SHC and applies to both Brownian motions and jump processes. This is a joint work with Kei Kobayashi (Fordham University).

 Abstract: Gundy and Varopoulos introduced the probabilistic representation of singular integrals and Fourier multipliers such as Hilbert transforms and Riesz transforms as conditional expectations of some stochastic integrals. Combining with the sharp martingale inequalities by Burkholder and Banuelos-Wang, the representations have played a crucial role in finding the sharp, or nearly sharp, $L^p$-bounds for these operators in a variety of geometric settings. Motivated by a recent breakthrough of Banuelos and Kwasnicki on the sharp $\ell^p$-norm of the discrete Hilbert transform, we construct a natural collection of discrete operators on $\mathbb{Z}^d$  which have $\ell^p$-norms independent of the dimension. This collection of discrete operators include the probabilistic discrete Riesz transforms, which are the analogues of the probabilistic discrete Hilbert transform used in the paper by Banuelos-Kwasnicki. In this talk, we discuss the construction of the probabilistic discrete operators, their $\ell^p$ bounds, and related open problems. This is based on joint work with Rodrigo Banuelos and Mateusz Kwasnicki.

 Abstract: The ellipitic Ginibre ensemble belongs to a random matrix ensemble which interpolates between the Ginibre and Gaussian orthogonal ensembles. A common characteristic of the matrix models is a presence of the real eigenvalues whose distributions depend on the non-Hermiticity parameter. In this talk, we consider the real eigenvalue densities of the elliptic Ginibre ensembles for two different regimes which are called strong and weak non-Hermiticity regimes. We present the finite size corrections for the densities for the both regimes. This is based on a joint work with Sung-Soo Byun.

 Abstract: We consider a system of N neurons, whose membrane potentials follow a mean-field interaction dynamics. More precisely, for each neuron, this potential decreases at constant rate and, on the other hand, it is set back to 0 when the neuron emits a spike, which also induces an increase of the potential of all the other neurons. The spikes occur at random times, at a rate lambda(u) which depends on the membrane potential u. When lambda(u) is zero at zero and differentiable then, for all N, the system almost surely stops in finite time, i.e. there is only a finite number of spikes, followed by a deterministic decay of the system toward 0. We will see that, under some conditions, however, the system is metastable, in the sense that the following points hold: 1) the non-linear limit (as N->infinity) system converges to a unique non-zero equilibrium ; 2) the extinction time of a finite system of size N is exponentially large in N ; 3) the average potential of the system fastly approaches a constant positive value, and the exit times from neighborhood of this value converge (as N->infinity) to the exponential law. The proofs rely on coupling techniques. This is a joint work with Eva Löcherbach.

 Abstract: Stochastic dynamical systems often exhibit some form of local stability (metastability) and phase transition from a domain of local stability (a metastable set) to another domain of local stability. For example, under the presence of multiple modes in the training landscape, stochastic gradient descent (SGD) can be attracted to and “stabilizes” at a suboptimal local minimum for a long time, even if its true stationary distribution has little mass at the local minimum and is predominantly concentrated elsewhere. Such phenomena has a close connection to the large deviations of small-noise dynamical systems: If one considers a finite time horizon, escaping from a local minimum is a large deviation event with small probability. On the other hand, if one waits for a long enough time, the SGD is guaranteed to escape any local minima and eventually find the global minimum. In this talk, we propose a new local uniform version of the heavy-tailed large deviations formulation and introduce the notion of asymptotic atom. We then show that one can transform such large deviations formulation to a sharp local stability analysis (i.e., convergence of exit time and location distributions). These machineries provide a streamlined framework for the heavy-tailed counterpart of the classical Freidlin-Wentzel theory and Eyring-Kramers formula which have been limited to the light-tailed contexts. Moreover, we show that, due to the generality and precise nature of the asymptotics we develop, the analysis can be further elevated to a crisp characterization of the global dynamics of the stochastic systems. We illustrate this in the context of heavy-tailed SGDs and show that our framework successfully captures intricate mathematical structures that is unique to the heavy-tailed SGDs.

 Abstract: In this talk I will outline the recent result together with Vesa Julin and Lauri Viitasaari concerning a characterization of the Eyring-Kramers formula that works also at degenerate saddle points, without resorting to Taylor expansions. I will explain how we used the ideas from geometric function theory to prove a very general capacity estimate.

 Abstract: In this talk, I will study the so-called cutoff phenomenon for the Langevin dynamics with a strongly coercive potential and driven by an additive noise with small amplitude. When the driven noise is the Brownian motion, I will show that the total variation distance between the current state and its equilibrium distribution decays around the mixing time from its maximum to zero abruptly. This is known in the literature as the cutoff phenomenon introduced in the context of card shuffling. When the noise is alpha-stable or more general Layered stable with index alpha>3/2, cutoff phenomenon still holds whereas for alpha\leq 3/2 the coupling techniques do not apply and hence we cannot conclude if the cutoff phenomenon still holds.