台大校區機率演講公告Probability Seminars on NTU Campus

I will review recent progress in joint work with Hubert Lacoin (IMPA) about the regularity of the phase transition of the directed polymer model in spatial dimensions $d\\geq 3$. This model describes a random path (called polymer in this context) affected by a random environment of i.i.d. weights and has received much attention recently because of its (partly conjectural) relationship to the KPZ universality class. It is known that in dimensions $d\\geq 3$, the model undergoes a phase transition from a weak disorder, high temperature phase (where the polymer converge to Brownian motion under the usual scaling) to a strong disorder, low temperature phase where the polymer localize into small, favorable regions and are expected to scale superdiffusively. This phase transition is characterized through the positivity (or not) of an associated martingale. In a number of works, we have analyzed the behavior of this model close to its critical point. We have shown that beyond the critical value $\\beta_c$ for the disorder intensity beta, the associated martingale converges to zero exponentially fast at a rate $f(\\beta)$. The main result of this talk is that we obtain upper and lower bounds on $f(\\beta_c+u)$ as $u \\to 0$.

In this talk, I will discuss a recent result concerning the complexity of an anisotropic Gaussian field coupled with a non-rotationally invariant potential arising in spin glass theory. We derive a variational formula for the logarithmic asymptotics of the expected number of critical points. As an application, by identifying the critical point of the associated phase transition, we obtain an upper bound on the ground state energy of the model. This is a joint work with Wei-Kuo Chen and Arnab Sen.

We consider the directed polymer model on $\\mathbb{Z}^d$, focusing on the critical dimension $d = 2$. Our main contribution is to give a sharp lower bound on the free energy in the high-temperature regime. Our proof uses a percolation argument inspired by Lacoin (2010), for which we introduce a key property of bounded \"log-energy\": this property quantifies the regularity of the polymer measures at diffusive scales and we show that it propagates along open paths. Combined with Theorem 2.8 of Berger, Caravenna, and Turchi (2025), this gives upper and lower bounds on the free energy that are matching up to a multiplicative constant.

1. Introduction & Purposes

The primary goal of this lecture series is to introduce several emerging topics in evolutionary biology. In particular, we will focus on the NK model, introduced by Kauffman, Levin, and Weinberger, which is a random fitness function defined on the genomes of certain species with N genetic loci, each interacting with K others. This model has broad applicability in the study of evolution and natural selection, as it captures the inherent ruggedness of fitness landscapes.

2. Outline & Descriptions

Throughout the series, we will highlight recent developments, outline the underlying mathematical approaches, and present open problems that are especially well suited for graduate students and junior researchers seeking to broaden their research directions.

3. Prerequisites

Participants are expected to be familiar with material from at least a first-semester graduate course in probability.

4. Registration

https://forms.gle/D9XjevbBKhbd71Fh8

While the space of rooted gaphs admits a natural (metrizable) topology, this is not the case of the space of unrooted graphs. This problem can be overcome by turning a finite graph into a probability measure on the space of rooted graphs: simply choose the root uniformly at random. This simple but fruitful idea allows to define the Benjamini-Schramm convergence of a sequence of finite graphs as the weak-* limit of the associated probability measures. Any such limit is automatically a unimodular measure on the space of rooted graphs.
With my colleagues we computed the Benjamini-Schramm limit for Rauzy graphs associated to a subshift. For subshifts of finite type, the limit is the pushforward of the unique measure of maximal entropy. This measure is also characterised by its support $S$. More precisely, the limit of Rauzy graphs is the unique unimodular measure on $S$.

This is joint work with R. Grigorchuk, T. Nagnibeda, A. Skripchenko and G. Veprev

Scaling limits of random walks are a widely studied topic in probability theory, classically connected with diffusion and the functional central limit theorem. Originally studied in the context of analytic number theory, in the last 30-40 years $p$-adic state spaces have been investigated as analytically tractable settings in which to model hierarchical and ultrametric phenomena. By mimicking the definition of the fractional Laplace operator, it is possible to define certain radially-symmetric Levy processes valued in these state spaces. Recent work has shown that these Levy processes can be obtained as scaling limits of certain long-range random walks.

In these lectures, we will assume no background beyond a standard course in real analysis and general familiarity with probability theory. We will start with an introduction to the classical study of random walks and their scaling limits, including discussion of certain technical difficulties which arise when working with continuous-time processes. We will also introduce the notion of a scaling limit and, from the functional perspective of \"functions as points,\" motivate the concept of weak convergence in Skorokhod space. We will then define the state space of $p$-adic numbers from a concrete, analytic approach, requiring no familiarity with number theory beyond the existence and uniqueness of prime factorizations. We will introduce the theory of integration in this setting, the Fourier transform and an analogue of the fractional Laplace operator, and use these to define a class of Levy processes. We will then outline recent work exhibiting these processes as scaling limits of random walks. Time permitting, we will touch on ongoing work with certain processes which are not radially symmetric.

Can one emulate the simple random walk without actually doing anything random? This talk will be about a deterministic version of random walk called rotor walk, and we will measure its performance in emulating the simple random walk with respect to different parameters, e.g., the shape of the trajectory, number of returns to the origin, etc. In particular, we will see that the number of returns to the origin for the rotor walk can be made equal to the same number for the simple random walk. This resolves a conjecture of Florescu, Ganguly, Levine, and Peres (2014). Parts of this talk will be a joint work with Lila Greco, Lionel Levine, and Peter Li.

In this talk, we study a risk-sensitive portfolio optimization problem with a finite time horizon in a financial market consisting of one risk-free asset and one risky asset. The price dynamics of the risky asset are governed by an $\\alpha$-hypergeometric stochastic volatility model, where the volatility factor follows a positive solution of a linear stochastic differential equation. Using a dynamic programming approach, we derive the associated Hamilton--Jacobi--Bellman (HJB) equation. Furthermore, by applying a suitable Feynman--Kac representation, we establish the existence and uniqueness of a classical solution for the HJB equation. Finally, we prove a verification theorem and construct the optimal investment strategy.

Please refer to this attached pdf file for the abstract.