Research Seminar

 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: 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: 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.

 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: 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: 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: 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: 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: 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.

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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: 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: 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: 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: 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.