YMSC Probability Seminar 概率论讨论班

报告人 Speaker:Xiucai Ding (University of California, Davis)
组织者 Organizer:吴昊,杨帆,姜建平,顾陈琳
时间 Time:Monday,15:00-16:00, June 24, 2024
地点 Venue:Shuangqing Complex Building 双清综合楼

Upcoming talks:

SpeakerXiucai Ding 丁秀才, University of California, Davis

TimeMonday,15:00-16:00, June 24, 2024

VenueC548, Shuangqing Complex Building A 清华大学双清综合楼AC548

TitleManifold learning for noisy and high-dimensional datasets: challenges and some solutions

AbstractManifold learning theory has garnered considerable attention in the modeling of expansive biomedical datasets, showcasing its ability to capture data essence more effectively than traditional linear methodologies. Nevertheless, prevalent algorithms are primarily designed for low-dimensional and clean datasets, whereas contemporary biomedical datasets tend to be high-dimensional and noisy. This presentation addresses the adaptation of these algorithms to effectively accommodate the challenges posed by high dimensionality and noise in modern datasets.

Note: Jun Yin's talk on June 4 has been cancelled.

Title: Non-mean-field random matrices related to quantum chaos and Anderson conjecture

Speaker: 尹骏, 加州大学洛杉矶分校(Jun Yin, UCLA)

Time: Tuesday, 16:00-17:00, Jun 4, 2024 (cancelled)

Venue:C548, Shuangqing Complex Building A


The Quantum Chaos Conjecture has long captivated the scientific community, proposing a crucial spectral phase transition demarcating integrable systems from chaotic systems in quantum mechanics. In integrable systems, eigenvectors typically exhibit localization with local eigenvalue statistics adhering to the Poisson distribution. In contrast, chaotic systems are characterized by delocalized eigenvectors, and their local eigenvalue statistics reflect the Sine kernel distribution reminiscent of the conventional random matrix ensembles GOE/GUE. Similarly, the Anderson conjecture reveals comparable phenomena in the context of disordered systems. This talk delves into the heart of this phenomenon, presenting a novel approach through the lens of random matrix models. By utilizing these models we aim to provide a clear and intuitive demonstration of the same phenomenon shedding light on the intricacies of these long-standing conjectures.


Past talks:

Speaker:黄骄阳, 宾夕法尼亚大学 (Jiaoyang Huang, University of Pennsylvania)
Time: Thursday, 16:00-17:00, Jun 13, 2024
Venue:C548, Shuangqing Complex Building A

Title:Extreme eigenvalues of random regular graphs


Extremal eigenvalues of graphs are of particular interest in theoretical computer science and combinatorics. Specifically, the spectral gap—the gap between the first and second largest eigenvalues—measures the expanding property of the graph. In this talk, I will focus on random $d$-regular graphs, for which the largest eigenvalue is $d$.

I'll first explain some conjectures on the extremal eigenvalue distributions of adjacency matrices of random $d$-regular graphs. In the second part of the talk, I will discuss a new proof of Alon's second eigenvalue conjecture, which asserts that with high probability, the second eigenvalue of a random $d$-regular graph concentrates around $2\sqrt{d-1}$. Our proof shows that the fluctuations of these extreme eigenvalues are bounded by $N^{−2/3+\varepsilon}$, where $\varepsilon>0$ can be arbitrarily small. This gives the same order of fluctuation as the eigenvalues of matrices from the Gaussian Orthogonal Ensemble. This work is based on joint research with Theo McKenzie and Horng-Tzer Yau.


Title: Skew Brownian flow: a new approach

Speaker: 王诚石,复旦大学 (Chengshi Wang, Fudan University)

Time: Thursday, 16:00-17:00, May 23, 2024

Venue: C548, Shuangqing Complex Building A


The skew Brownian motion is constructed by assigning signs to Brownian excursions away from 0, each excursion being positive with probability p and negative with probability 1-p. It can equivalently (Harrison-Shepp, 1981) be seen as the strong solution of the SDE dX_t=dB_t + (2p-1) dL_t(X) where L_t(X) denotes the local time of the diffusion at 0. The skew Brownian flow as studied by Burdzy-Chen (2001) and Burdzy-Kaspi (2004) is the flow of solutions driven by the same Brownian motion but starting from any time-space point in the plane. In this talk, I will give results on the properties of meeting level processes and the exact Hausdorff dimension of bifurcation times of different types appeared in the skew Brownian flow. If time permits, I will introduce another coalescing stochastic flow, called the BESQ flow, as the key to those problems. Joint work with Elie Aïdékon and Yaolin Yu.

Title: Free energy in non-convex mean-field spin glass models

Speaker: 陈鸿宾,法国高等研究院(Hong-Bin Chen, IHES)

Time: Tuesday, 16:00-17:00, May 21, 2024

Venue: B725, Shuangqing Complex Building A

We start by reviewing the classical Sherrington-Kirkpatrick (SK) model. In this model, +1/-1-valued spins interact with each other subject to random coupling constants. The covariance of the random interaction is quadratic in terms of spin overlaps. Parisi proposed the celebrated variational formula for the limit of free energy of the SK model in the 80s, which was later rigorously verified in the works by Guerra and Talagrand. This formula has been generalized in various settings, for instance, to vector-valued spins, by Panchenko. However, in these cases, the convexity of the interaction is crucial. In general, the limit of free energy in non-convex models is not known and we do not have variational formulas as valid candidates. Here, we report recent progress through the lens of the Hamilton-Jacobi equation. Under the assumption that the limit of free energy exists, we show that the value of the limit is prescribed by a characteristic line; and the limit (as a function) satisfies an infinite-dimensional Hamilton-Jacobi equation "almost everywhere". This talk is based on a joint work with Jean-Christophe Mourrat.

Title: Speed of Random Walks in Dirichlet Environment on a Galton-Watson Tree

Speaker:Xiao Yang 肖阳,Qian Dongjian 钱东箭 (复旦大学)
Time:4:00-5:00 pm, April 18, 2024
Venue:Shuangqing Complex Building 双清综合楼C548


This paper mainly deals with a transient random walk in Dirichlet Environment, or equivalently a linearly edge reinforced random walk, on a Galton-Watson tree. We compute stationary distributions of the environment seen from the particle in both quenched and annealed cases. With these stationary measures, we provide the almost necessary and sufficient criteria for positive speed and give a formula for speed when it is positive.

Title: The scaling limit of the volume of loop-O(n) quadrangulations

Speaker:Hu Xingjian 胡行健 (复旦大学)
Time:4:00-5:00 pm, April 11, 2024
Venue:Shuangqing Complex Building 双清综合楼C548


In this talk we will discuss the O(n) loop model on quadrangulations. We will first introduce the model, its basic properties and several related work for preliminaries. Then we discuss the volume of maps and state our scaling limit result for the volume of the O(n) quadrangulations. In the end, we will state the outline of our proof and some observations of the relationship with branching random walk. This talk is based on joint work with Élie Aïdékon (Fudan University) and William Da Silva (University of Vienna).

Title: Biased random walk on dynamical percolation

Speaker:Dominik Schmid (University of Bonn)

Time:4:00-5:00 pm, March 7, 2024

Venue:Shuangqing Complex Building 双清综合楼C548


We consider a biased random walk on dynamical percolation and discuss the existence and the properties of the linear speed as a function of the bias. In particular, we establish a simple criterion to decide whether the speed is increasing or decreasing for large bias. This talk is based on joint work with Sebastian Andres, Nina Gantert, and Perla Sousi.

Title: Spectral gap and two point functions for spin glass models

Speaker:Changji Xu (Harvard University)

Time:4:00-5:00 pm, Dec. 21, 2023

Venue:Shuangqing Complex Building 双清综合楼C548


In this talk, I will discuss recent progress in mean-field spin glass models. Specifically, I will present a method to prove that the spectral gap of the Dirichlet form associated with Glauber dynamics is of order one at sufficiently high temperatures. Furthermore, I will talk about estimates on two-point functions for the SK model under high-temperature conditions.

Title: Universal Rank Inference via Residual Subsampling with Application to Large Networks

Speaker:Han Xiao (韩潇, 中国科学技术大学)

Time:4:00-5:00 pm, Dec. 7, 2023

Venue:Shuangqing Complex Building 双清综合楼C548


Determining the precise rank is an important problem in many large-scale applications with matrix data exploiting low-rank plus noise models. In this paper, we suggest a universal approach to rank inference via residual subsampling (RIRS) for testing and estimating rank in a wide family of models, including many popularly used network models such as the degree corrected mixed membership model as a special case. Our procedure constructs a test statistic via subsampling entries of the residual matrix after extracting the spiked components. The test statistic converges in distribution to the standard normal under the null hypothesis, and diverges to infinity with asymptotic probability one under the alternative hypothesis. The effectiveness of RIRS procedure is justified theoretically, utilizing the asymptotic expansions of eigenvectors and eigenvalues for large random matrices recently developed. The advantages of the newly suggested procedure are demonstrated through several simulation and real data examples.


Title: Universality of extreme eigenvalues of a large non-Hermitian random matrix 

Speaker:Xu Yuanyuan (许媛媛,中国科学院数学与系统科学研究院)

Time:4:00-5:00 pm, Nov. 30, 2023

Venue:Shuangqing Complex Building C548


We will report on recent progress regarding the universality of the extreme eigenvalues of a large random matrix with i.i.d. entries. Beyond the radius of the celebrated circular law, we will establish a precise three-term asymptotic expansion for the largest eigenvalue (in modulus) with an optimal error term. Based on this result, we will further show that the properly normalized largest eigenvalue converges to a Gumbel distribution as the dimension goes to infinity. Similar results also apply to the rightmost eigenvalue of the matrix. These results are based on several joint works with Giorgio Cipolloni, Laszlo Erdos, and Dominik Schroder.


Title: On the influence of edges in first-passage percolation on Z^d

Speaker:Ron Peled (Tel Aviv University, Princeton University and the Institute for Advanced Study )
Time:10:00-11:00 am, Nov. 30, 2023

Venue:Shuangqing Complex Building C548


First-passage percolation is the study of the geometry obtained from a random perturbation of Euclidean geometry. In the discrete setting, one assigns random, independent and identically distributed, lengths to the edges of the lattice Z^d and studies the resulting geodesics - paths of minimal length between points. Given eps>0 and v in Z^d, which edges have probability at least eps to lie on the geodesic between the origin and v? It is expected that all such edges lie at distance at most some r(eps) from either the origin or v, but this remains open in dimensions d>=3. We establish the closely-related fact that the number of such edges is at most some C(eps), uniformly in v. We further provide quantitative bounds for eps tending to zero as ||v|| tends to infinity. This addresses a problem raised by Benjamini–Kalai–Schramm (2003). Joint work with Barbara Dembin and Dor Elboim.

Title: A new correlation inequality for Ising models with external fields

Speaker:Song Jian (宋健,山东大学)

Time:4:00-5:00 pm, Nov. 23, 2023

Venue:Shuangqing Complex Building C548


We study ferromagnetic Ising models on finite graphs with an inhomogeneous external field, where a subset of vertices is designated as the boundary. We show that the influence of boundary conditions on any given spin is maximised when the external field is identically 0. One corollary is that spin–spin correlations are maximised when the external field vanishes and the boundary condition is free, which proves a conjecture of Shlosman. In particular, the random field Ising model on Z^d, d ≥ 3, exhibits exponential decay of correlations in the entire high temperature regime of the pure Ising model. Another corollary is that the pure Ising model in d ≥ 3 satisfies the conjectured strong spatial mixing property in the entire high temperature regime.


Title:One-arm exponent of critical metric graph Gaussian free field in high dimensions

Speaker:Cai Zhenhao 蔡振豪

Time:Thur.,16:00-17:00,Nov. 9,2023

Venue:Shuangqing C548



We prove that for the critical level-set of Gaussian free field on the metric graph generated by Z^d (d>6), the one-arm probability (i.e. the probability of the event that the origin is connected to the boundary of the box B(N)) is proportional to N^{-2}, where B(N) is centered at the origin and has side length 2N. This paper is a joint work with Prof. Jian Ding. In addition to presenting the proof idea, this talk will also involve some interesting open questions, including the one-arm exponent in low dimensions, the structure of large clusters, and the incipient infinite cluster (IIC).



I am a PhD student at the School of Mathematical Science, Peking University. My supervisor is Prof. Yuan Zhang. Currently, my research interests include some stochastic models related to random walks, such as random interlacements, Gaussian free field, and diffusion-limited aggregation.



题目: Stochastic equations with Divergence-free Drifts and its application




摘要:This talk focuses on time-inhomogeneous stochastic equations driven by additive noises with divergence-free drifts in Lorentz space $L^{\infty}(0,T;L^{d,\infty}(\mR^d))$. We will cover conditional and unconditional well-posedness, along with Aronson type estimates, followed by practical applications to the McKean-Vlasov equation with singular interaction kernels.



Speaker:Shangjie Yang(Bar-Ilan University)

Time:Thur., 10:00-11:00 am, Nov. 2, 2023

Venue: B725, Tsinghua University Shuangqing Complex Building A


Title: Phase Transitions in Quantum Many-body system 

Speaker:Wang Zhituo 王智拓

Time:Oct. 26th 4pm-5pm

Venue:Shuangqing Complex Building 双清综合楼C548



In this talk I will present some recent progress on the construction of ground state of the honeycomb Hubbard model, which is a prototypical model for studying phase transitions in quantum many-body system. Using fermionic cluster expansions and constructive renormalization theory, we proved that the ground state of this model is not a Fermi liquid in the mathematical precise sense of Salmhofer. We also derived the non-perturbative critical temperature for the phase transition.

Title:High-dimensional canonical correlation analysis 

Speaker:Vadim Gorin(UC Berkeley)

Time:Oct. 25th 4pm-5pm
Venue: Shuangqing Complex Building 双清综合楼C654



Canonical correlations have two faces: from one side, in statistics they give a way to measure dependence between two datasets. From the other side, in linear algebra they represent a canonical form to which a pair of two linear subspaces can be transformed. In the talk we will discuss the theory of canonical correlations when the dimensions of the subspaces are large, in the setting related to both random matrix theory and high-dimensional statistics.


题目: Stationary fluctuations for the facilitated exclusion process






We consider the Facilitated Exclusion Process (FEP) in one dimension, which is an interacting particle system with degenerate rates. In this model, a particle could jump if and only if there is one particle in its neighbors and no particle at the target site. We derive stationary fluctuations for the FEP in the symmetric, weakly asymmetric and asymmetric cases.  In particular, we observe a phase transition from the Edwards-Wilkinson universality to the Kardar-Parisi-Zhang universality. This is based on joint work with Clément Erignoux. We shall also discuss stationary fluctuations for interacting particle systems with two or more conservation laws, where a new universality class, namely the Fibonacci universality class, has been predicted by physicists.








主讲人:刘全升(法国 Universite Bretagne-Sud)

课程摘要:随机矩阵的乘积及其应用,包括在随机环境中的分枝过程,分枝随机游动,和 Mandelbrot 乘积瀑布方面的应用。

课程时间:8 月 7 日、8 月 9 日、8 月 14 日上午 10:00-11:30

课程地点:静斋 105



Three lectures are planned to present some recent limit theorems for products of independent and identically distributed random matrices, and related topics. We focus on precise large deviations and convergence rates in the Gaussian approximation, and applications to multi-type branching processes in random environments, branching random walks, and Mandelbrot’s cascades.



Quansheng Liu is professor of exceptional class at University of South Brittany (Univ. Bretagne-Sud, France). From 2007 to 2013 and from 2017 to 2022, he has been the Director of the Laboratory of Mathematics of University of South Brittany, and Associate Director of the Laboratory of Mathematics of Atlantic Brittany. He received his B.S. and M.S. degrees in Mathematics from Wuhan University respectively in 1984 and 1987, and his Ph.D. degree in Probability from University Paris 6 in 1993. His research interests include probability theory and image processing. He has published over 100 research articles in scientific journals, such as J. Eur. Math. Soc., Prob. Th. Rel. Fields, Ann. Prob., Ann. Appl. Prob., IEEE Trans. Image Processing, SIAM J. Imaging Sciences, and J. Sci. Comput. , etc.

Title:Strong noise limit of stochastic differential equations

Speaker:Joseph Najnudel (University of Bristol)

Time:Mon., 2:00-3:00 pm, July 3, 2023

Venue:Ning Zhai W11



In this talk, we study some stochastic differential equations related to continuous measurements of open quantum systems. In these equations, we consider the regime where the coefficient on the noise tends to infinity. Under some conditions, we get pure jump processes which correspond to continuous time Markov chains, these processes being decorated with spikes distributed like a Poisson point process. The proof of this result, which needs some work to be rigorously stated, uses the behaviour of the Brownian local time as one of its main tools. This is a joint work with Benoist, Bernardin, Chetrite, Chhaibi and Pellegrini.



Joseph Najnudel is an Associate Professor at the University of Bristol. His research interests are random matrix theory and stochastic analysis.


题目: Extreme gap problems for random matrices


报告人:姚东,江苏师范大学数学研究院副教授。2021年9月博士毕业于美国杜克大学,师从Rick Durrett教授;2017年7月本科毕业于中国科学技术大学数学与应用数学专业。



In the talk, I will give a brief review of known results on the extreme gap problems (smallest and largest gaps of the eigenvalues) of various random matrix ensembles. Then I will present our recent work about the smallest gap of the Gaussian symplectic ensemble. This completes the picture of the small gap problem of classical Gaussian β ensembles for β=1, 2, 4. Our analysis can potentially be applied to other classical ensembles with a Pfaffian structure.


Title: Modulation and amplitude equations on bounded domains for nonlinear SPDEs driven by cylindrical α-stable Lévy processes

Time: 3:00-4:00 pm, May 24, 2023

Speaker: 袁胜兰(Institut für Mathematik, Universität Augsburg)

Venue: Ning Zhai W11



In the present work, we establish the approximation via modulation or amplitude equations of nonlinear stochastic partial differential equation (SPDE) driven by cylindrical α-stable Lévy processes. We study SPDEs with a cubic nonlinearity, where the deterministic equation is close to a change of stability of the trivial solution. The natural separation of time scales close to this bifurcation allows us to obtain an amplitude equation describing the essential dynamics of the bifurcating pattern, thus reducing the original infinite dimensional dynamics to a simpler finite-dimensional effective dynamics. In the presence of a multiplicative stable Lévy noise that preserves the constant trivial solution we study the impact of noise on the approximation. In contrast to Gaussian noise, where non-dominant pattern are uniformly small in time due to averaging effects, large jumps in the Lévy noise might lead to large error terms, and thus new estimates are needed to take this into account. This is the joint work with Dirk Blömker.


Title: Gaussian inequalities and multiplicative chaos

Speaker: Yichao Huang 黄逸超 (Beijing Institute of Technology)

Time: 2:00-3:00 pm, May 24, 2023

Venue:Ning Zhai W11



Many crucial properties from the theory of Gaussian multiplicative chaos (initiated by Kahane in the 80s’) are based on various Gaussian comparison or decorrelation inequalities. We review some basic ingredients (with details) and give several recent applications and generalizations.


Title: The derivative martingale in a branching Lévy process


Time:Wed., 14:00-15:00, May 17, 2023

Venue:Ning Zhai W11


In the study of the branching Brownian motion, the convergence of the derivative martingale is of significant interest, since the limit can be used to study the travelling wave solutions of the FKPP (Fisher–-Kolmogorov–-Petrovskii–-Piskunov) equation.

Recently, Bertoin introduces branching Lévy processes, generalizing the branching Brownian motion to a very general class of branching particle systems. For a branching Lévy process, we obtain a necessary and sufficient condition for the convergence of the derivative martingale to a non-trivial limit. This extends previously known results for branching Brownian motions and branching random walks.

Joint work with Bastien Mallein.




Title:Factor Modeling for Clustering High-dimensional Time Series

Speaker: 张博,中国科学技术大学

Time: Wed., 13:30-14:30, May 10, 2023

Venue: Ning Zhai W11



We propose a new unsupervised learning method for clustering a large number of time series based on a latent factor structure. Each cluster is characterized by its own cluster-specific factors in addition to some common factors which impact on all the time series concerned. Our setting also offers the flexibility that some time series may not belong to any clusters. The consistency with explicit convergence rates is established for the estimation of the common factors, the cluster-specific factors, and the latent clusters. Numerical illustration with both simulated data as well as a real data example is also reported. As a spin-off, the proposed new approach also advances significantly the statistical inference for the factor model of Lam and Yao (2012).





Title: Improved Bounds for Sampling Solutions of Random CNF Formulas

Speaker: 杨宽(上海交通大学)

Time: Wed., 14:00-15:00, April 26, 2023

Venue:Ning Zhai W11



Let Φ be a random k-CNF formula on n variables and m clauses, where each clause is a disjunction of k literals chosen independently and uniformly. Our goal is, for most Φ, to (approximately) uniformly sample from its solution space.

Let α=m/n be the density. The previous best algorithm runs in time n^poly(k,α) for any α≲2^(k/300) [Galanis, Goldberg, Guo, and Yang, SIAM J. Comput.'21]. In contrast, our algorithm runs in near-linear time for any α≲2^(k/3). This work was published in SODA'23.



Title:Mean-field interacting particle systems and McKean-Vlasov equation






McKean-Vlasov quation describes heat diffusion in porous media and is the limit of propagation of chaos of system of interacting particle systems of mean field with a great number N of particles, known in 80s. Carrillo-McCann-Villani(03) renewed the subject by regarding MV equation as the gradient flow of the entropy functional on the space of probability measures w.r.t. the Otto- Villani's differential calculus, and proved the strict convexity of the entropy and so the entropical exponential convergence of MV equation by assuming the convexity of the confinement potential and the interacting potential. Their convexity assumption excludes the non-convex models which may have phase transition at low temperature, such as Curie-Weiss.

In this talk I will present some recent generalizations covering the non-convex cases on this subject, by using the tools such as Poincare inequality, log-Sobolev inequality, gradient estimate for interacting particle system. This talk is based on a series of joint works with A.Guillin, W. Liu, C. Zhang.



Title:Variational formulas for asymptotic variance of general Markov processes


Time:Wed., 14:00-15:00, Mar.15, 2023




The asymptotic variance is an important criterion to evaluate the performance of Markov processes, especially for the central limit theorems. In this talk, we give a variational formula for the asymptotic variance of (nonreversible) Markov processes. The variational formula provides many applications, extending the classical Peskun’s comparison theorem to non-reversible Markov processes, and obtaining several comparison theorems between Markov processes with various perturbations. Based on joint work with Yong-Hua Mao.






Title:  Renormalization and quantitative homogenization

Speaker:Scott Armstrong (NYU)

Time:Thur., 17:00-18:00, Dec.1,2022

Venue:Zoom Meeting ID: 271 534 5558;Passcode: YMSC


I will present an overview of the theory of quantitative homogenization for elliptic and parabolic equations, with an emphasis on the "coarse-graining" point of view. I will begin by trying to motivate the results with some applications to models in mathematical physics. Time permitting, I will also discuss some open problems.



Scott Armstrong is Professor of Mathematics at the Courant Institute of Mathematical Sciences at NYU. His general research interests lie in partial differential equations, the calculus of variations and probability theory. His focus for the last several years has been in homogenization of PDEs in random media and related problems in probability and statistical mechanics.




Title: Random conductance models with stable-like long range jumps

Speaker: Chen Xin (SJTU), Wang Jian (Fujian Normal University)

Time: Wed.,9:50-11:30am, Nov.30,2022

Venue: Tencent Meeting ID: 355 023 8301


In this talk, we consider random conductance models with stable-like long range jumps, and obtain the quenched invariance principle (QIP) and  a quantitative version of stochastic homogenization for the scaled random walks with explicit polynomial rates up to logarithmic corrections.For QIP, we utilize probabilistic potential theory for the corresponding jump processes, and two essential ingredients of our proof are the tightness estimate and the Hölder regularity of caloric functions for non-elliptic stable-like processes on graphs.On the other hand, the proof of quantitative homogenization result is based on energy estimates of the localized corrector and multi-scale Poincarein equalities for non-local Dirichlet forms.



1、陈昕,上海交通大学数学科学学院教授,国家优秀青年基金获得者。 主要从事与随机分析及相关问题的研究,包括泛函不等式,跳过程的随机分析,流形上的随机分析等等。

2、王健,福建师范大学数学与统计学院教授、博士生导师、国家杰出青年基金获得者。 主要从事随机过程与随机分析方向的研究,特别是Lévy型过程的随机分析。




时间:2022-11-10 周四 15:00-16:00



标题:On Natural Measures of Several Random Fractals Induced from SLE and CLE
摘要:Schramm-Loewner Evolution(SLE), together with its loop version, namely Conformal Loop Ensemble(CLE), successfully describes the scaling limits of interfaces of many 2d statistical physics models. This talk focus on natural measures defined on the SLE- and CLE-related random fractals, namely the cut points and boundary touching points of SLE as well as the pivotal points and the carpet/gasket of CLE. We show that several simple axioms can uniquely characterize these SLE- and CLE-related natural measures. Based on a joint work with Xinyi Li.