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

In this lecture, we'll review some recent results regarding spectral gaps and logarithmic Sobolev inequality for Glauber dynamics of mean-field spin glass models. In particular, we will present a method to prove that the spectral gap of the Glauber dynamics is of order one at sufficiently high temperature. In addition, we will review certain estimates on two point functions for the SK model satisfying a modified AT condition.

Within the province of condensed matter physics, there exists a variety of interesting physical phenomena where we are concerned with the statistical fluctuations exhibited by an essentially linear elastic object, such as a hydrophilic polymer chain wafting in water. Due to the thermal fluctuation, the shape of the polymer chain should be understood as a random path. The water in this physical system plays the role of the disordered environment that contains randomly placed hydrophobic molecules as impurities, which repel the hydrophilic monomers that the polymer chain consists of. This physical system is called the directed polymer model. The major problem about this system is to investigate the behavior of the polymer chain for various disorder strengths.

In this talk, we will focus on the $(d+1)$-dimensional directed polymer model when $d \\geq 3$. In this case, the system is quite sensitive to the disorder strength. Moreover, it is widely believed that the model has a phase transition when the disorder strength is strong enough. Consequently, it is crucial to investigate the behavior of system until a critical disorder strength that the system may have a phase transition. As a result, the goal of this talk is to present the results given by D. Lygkonis and N. Zygouras [1], which studied the limiting fluctuations of the partition function and the free energy of the directed polymer model when the disorder strength is less than a critical value.

[1] D. Lygkonis and N. Zygouras. Edwards–Wilkinson fluctuations for the directed polymer in the full $L^2$-regime for dimensions $d \\geq 3$. <i>Annales de l’Institut Henri Poincaré</i>, Probabilités et Statistiques, 58(1):65 – 104, 2022. doi:10.1214/21-AIHP1173.

Within the province of condensed matter physics, there exists a variety of interesting physical phenomena where we are concerned with the statistical fluctuations exhibited by an essentially linear elastic object, such as a hydrophilic polymer chain wafting in water. Due to the thermal fluctuation, the shape of the polymer chain should be understood as a random path. The water in this physical system plays the role of the disordered environment that contains randomly placed hydrophobic molecules as impurities, which repel the hydrophilic monomers that the polymer chain consists of. This physical system is called the directed polymer model. The major problem about this system is to investigate the behavior of the polymer chain for various disorder strengths.

In this talk, we will focus on the $(d+1)$-dimensional directed polymer model when $d \\geq 3$. In this case, the system is quite sensitive to the disorder strength. Moreover, it is widely believed that the model has a phase transition when the disorder strength is strong enough. Consequently, it is crucial to investigate the behavior of system until a critical disorder strength that the system may have a phase transition. As a result, the goal of this talk is to present the results given by D. Lygkonis and N. Zygouras [1], which studied the limiting fluctuations of the partition function and the free energy of the directed polymer model when the disorder strength is less than a critical value.

[1] D. Lygkonis and N. Zygouras. Edwards–Wilkinson fluctuations for the directed polymer in the full $L^2$-regime for dimensions $d \\geq 3$. <i>Annales de l’Institut Henri Poincaré</i>, Probabilités et Statistiques, 58(1):65 – 104, 2022. doi:10.1214/21-AIHP1173.

The famous Gaussian Unitary Ensemble (GUE) consists of random Hermitian matrices with elements identically and independently (up to symmetry) distributed as Gaussian random variables. It also has the property that its probability measure is invariant under unitary conjugations. In general unitary invariant ensembles, one maintains the latter property but generalizes the measure by including a potential. We explain how the correlation functions of eigenvalues for such an ensemble can be compactly represented in terms of orthogonal polynomials for a weight involving the potential. Then we demonstrate how Deift, Kriecherbauer, McLaughlin, Venakides, and Zhou proved universality of the correlation functions both in the bulk of the spectrum and at the edge for general convex and analytic potentials, vastly generalizing known results for GUE. Their method uses a characterization of orthogonal polynomials in terms of the solution of a matrix Riemann-Hilbert problem, which applies for completely general weights. Finally, we discuss how the methodology was further extended by McLaughlin and the speaker to extend the universality results to potentials having just two Lipschitz continuous derivatives.

Phenomena of non-equilibrium growth processes are ubiquitous in nature. Many of these processes are believed to exhibit remarkably similar growth dynamics, which can be regarded as an interface evolves with time, changing its roughness while being subjected to random noise. In the original 1986 paper of M. Kardar, G. Parisi and Y.-C. Zhang [2], the authors predicted the dynamic of these processes, and indicated that the evolution can be described by the solution $\\mathscr{H}(x,t)$ of a stochastic PDE, which is called the Kardar-Parisi-Zhang (KPZ) equation. On the other hand, $\\mathscr{H}(x,t)$ plays an important role in statistical mechanics. Roughly speaking, through the Cole–Hopf transform $\\mathscr{H}(x,t) = \\log \\mathscr{U}(x,t)$, $\\mathscr{H}(x,t)$ can be regarded as the free energy of the continuous directed polymer model, which describes the behaviour of a hydrophilic polymer chain wafting in a disordered environment that contains randomly placed hydrophobic molecules as impurities. Here $\\mathscr{U}(x,t)$ plays the role of the partition function of this model, and solves a stochastic PDE, which is called the stochastic heat equation (SHE).

In this talk, we will focus on the SHE and KPZ equation when spatial dimension $d\\geq 3$. In this case, both of the SHE and KPZ equation are quite sensitive to $\\beta$, where $\\beta$ describes the strength of the white noise in these equations. Consequently, it is crucial to investigate the behavior of the SHE and KPZ equation for all $\\beta$. In a recent paper of F. Comets, C. Cosco, and C. Mukherjee [1], the authors proved the limiting fluctuation of $\\mathscr{H}(x,t)$ under a restriction on $\\beta$. Motivated by this result, in our recent work [3], we considered both $\\mathscr{H}(x,t)$ and $\\mathscr{U}(x,t)$, and established their limiting fluctuations in the entire $L^{2}$-regime (i.e., $\\beta < \\beta_{L^{2}}$). Here $\\beta_{L^{2}}$ is a critical value associated with the KPZ equation.

[1] F. Comets, C. Cosco, and C. Mukherjee. Space-time fluctuation of the Kardar-Parisi-Zhang equation in d $\\geq 3$ and the Gaussian free field. <i>arXiv preprint</i>, 2019. doi:10.48550/arXiv.1905.03200.

[2] M. Kardar, G. Parisi, and Y.-C. Zhang. Dynamic scaling of growing interfaces. <i>Phys. Rev. Lett.</i>, 56:889–892, 1986. doi:10.1103/PhysRevLett.56.889.

[3] T.-C. Wang. Space-time fluctuations for the SHE and KPZ equation in the entire $L^2$-regime for spatial dimensions $d \\geq 3$. in preparation. 2023+.

For a large class of second order elliptic differential operator $\mathcal{L}$, there is a related diffusion process X in which the operator $\mathcal{L}$ becomes an infinitesimal generator of X. The fundamental solution of heat equation of $\mathcal{L}$ is the transition density of the process X. In this talk, we will discuss theses relation of the general Lévy process through the Lévy Khintchine formula and its transition density.

It is widely conjectured that the Poisson-Dirichlet behavior appears universally in low-temperature disordered systems. However, this principle has been verified only for the particular models which are exactly solvable. In this talk, I will talk about the universal Poisson-Dirichlet behavior for the general log-correlated Gaussian fields. This is based on the joint work with Shirshendu Ganguly.

Stochastic processes are mathematical models of random phenomena and Lévy process is a large class of stochastic process. This course is an introduction to the theory of Lévy processes that covers definitions, infinitely divisibility and characteristic exponents. We also discuss examples of Lévy processes and Lévy Khintchine formula.

We consider the spread-out models of the self-avoiding walk and its finite-memory version, called the memory-$\tau$ walk. For both models, each step is uniformly distributed over the d-dimensional box $\{ x \in \mathbb{Z}^d : 0 < |x|_{\infty} \le L \}$. The critical point $p_c^{\tau}$ for the memory-$\tau$ walk is increasing in $\tau$ and converges to the critical point for the self-avoiding walk as $\tau$ goes to $\infty$. The speaker proved that the rate of convergence of $p_c^{\tau}$ in terms of $\tau$ is order of $\tau^{-(d-2)/2}$. Moreover, the speaker identified the exact expression of the coefficient of the dominant term of it. This improves the previous results obtained by Madras and Slade [Birkhäuser, The Self-Avoiding Walk, Lemma 6.8.6, 1993]. This talk is based on the speaker’s own work (http://arxiv.org/abs/2306.13936).

Stochastic processes are mathematical models of random phenomena and Lévy process is a large class of stochastic process. This course is an introduction to the theory of Lévy processes that covers definitions, infinitely divisibility and characteristic exponents. We also discuss examples of Lévy processes and Lévy Khintchine formula.

The quantum Ising model is a kind of model of ferromagnetic materials. According to quantum mechanics, ferromagnetism comes from a cooperation phenomenon of spins. The Hamiltonian operator (energy) for the quantum Ising model is defined by tensor products of Pauli matrices, which correspond to spins as a physical quantity. In this model, we impose a transverse field. The case without the transverse field is called the classical Ising model in particular. It is predicted that changing temperature or a transverse field causes phase transitions and critical phenomena from a point of view of numerical analysis (though this fact is mathematically proven in some special cases). It is well-known that critical exponents for the classical Ising model take the mean-field values in high dimensions. We are interested in whether or not the values are changed even when we impose the transverse field. In this series of talks, I will focus on the critical exponent $\gamma$ of the magnetic susceptibility and show that $\gamma = 1$ still holds for the quantum Ising model with the nearest-neighbor interaction. I will also mention an ongoing work about the lace expansion, which plays an important role in proving mean-field behavior for a lot of statistical mechanical models in high dimensions.

Directed Random Polymers model the motion of a random walk in a random potential. From its introduction into the mathematical community 35 years ago it has offered the ground for many interesting mathematical developments. In this talk I will review some of the milestones and point to some of the still tantalising open questions.

The quantum Ising model is a kind of model of ferromagnetic materials. According to quantum mechanics, ferromagnetism comes from a cooperation phenomenon of spins. The Hamiltonian operator (energy) for the quantum Ising model is defined by tensor products of Pauli matrices, which correspond to spins as a physical quantity. In this model, we impose a transverse field. The case without the transverse field is called the classical Ising model in particular. It is predicted that changing temperature or a transverse field causes phase transitions and critical phenomena from a point of view of numerical analysis (though this fact is mathematically proven in some special cases). It is well-known that critical exponents for the classical Ising model take the mean-field values in high dimensions. We are interested in whether or not the values are changed even when we impose the transverse field. In this series of talks, I will focus on the critical exponent $\gamma$ of the magnetic susceptibility and show that $\gamma = 1$ still holds for the quantum Ising model with the nearest-neighbor interaction. I will also mention an ongoing work about the lace expansion, which plays an important role in proving mean-field behavior for a lot of statistical mechanical models in high dimensions.

Is it true that if many elements of a group commute, then the group is commutative? What happens if most of the elements have order 2? Can every group be realised as the automorphism group of a \"nice\" graph? These questions have been solved in the 70-90's for finite groups. But what happens for infinite groups? How can we interpret \"many elements\" or \"most of the elements\" for infinite groups?

In this talk we will see how we can use random walks to solve the above questions for infinite (finitely generated) groups.

The talk is supposed to be a gentle introduction to random walks and infinite group theory; showing how probability theory and algebra nicely interact. No prerequisites are required and the talk will accessible to second year bachelor students.

In this talk, our lecture starts with introducing pre-Brownian motion which is defined from a Gaussian white noise. Going from pre-Brownian motion to Brownian motion requires the additional property of continuity of sample paths. Then we introduce some properties of Brownian sample paths, and establishes the strong Markov property.

The quantum Ising model is a kind of model of ferromagnetic materials. According to quantum mechanics, ferromagnetism comes from a cooperation phenomenon of spins. The Hamiltonian operator (energy) for the quantum Ising model is defined by tensor products of Pauli matrices, which correspond to spins as a physical quantity. In this model, we impose a transverse field. The case without the transverse field is called the classical Ising model in particular. It is predicted that changing temperature or a transverse field causes phase transitions and critical phenomena from a point of view of numerical analysis (though this fact is mathematically proven in some special cases). It is well-known that critical exponents for the classical Ising model take the mean-field values in high dimensions. We are interested in whether or not the values are changed even when we impose the transverse field. In this series of talks, I will focus on the critical exponent $\gamma$ of the magnetic susceptibility and show that $\gamma = 1$ still holds for the quantum Ising model with the nearest-neighbor interaction. I will also mention an ongoing work about the lace expansion, which plays an important role in proving mean-field behavior for a lot of statistical mechanical models in high dimensions.

In the first lecture we discussed extreme value theory for logarithmically correlated Gaussian fields. In this talk I will discuss what changes in the non-Gaussian Gaussian setup. A prime example is the study of cover time of certain planar graphs or two dimensional manifolds by random walk or Brownian motion. In spite of precise and beautiful links through isomorphism theorems, the question about the cover time of the 2D torus by a Wiener sausage (or its discrete analogue) requires new tools. I will describe some work, old and recent, on this question, culminating with limit theorems for the cover time. If time permits, I will briefly discuss other types of non Gaussian LCFs.

The discrete Gaussian free field (DGFF) and the simple random walk (SRW) have a close relationship via the generalized second Ray-Knight theorem, which is a distributional identity between the square of DGFF and the local time of SRW. Thanks to the theorem, we have witnessed rapid progress on the studies of the cover time (the first time at which SRW visits all the vertices) and thick points of SRW (sites frequently visited by SRW). In the first half of this lecture, I will state the generalized second Ray-Knight theorem and review results on the cover time due to Ding-Lee-Peres (2012) and Zhai (2018) where we can see beautiful applications of the theorem. In the second half, I will focus on applications to thick points of SRW.

The extreme value theory for Gaussian logarithmically correlated fields (G-LCFs) has emerged in the last decade as a powerful tool in the analysis of interface models, quantum gravity, random matrices and in a myriad of other applications.<br>The two dimensional Gaussian free field (and its discrete analogue) is an important motivating example of such a field. In this lecture, I will describe the relation and differences between the extreme value theory for i.i.d. variables and that for G.-LCFs, and introduce the relation with branching structures and various tools such as comparison theorems, scale decompositions and relations to branching random walks.

The discrete Gaussian free field (DGFF) is a centered Gaussian field on a graph whose covariance is given by the inverse of the graph Laplacian. It is a probabilistic model of interfaces and has connections with a lot of other models such as local times of random walks and branching random walks. In the first half of this lecture, I will give some motivation and basics of DGFF such as the random walk representation and the domain Markov property. In the second half, I will review some progress on the extreme value theory of DGFF on the integer lattice in three or higher dimensions.