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Insuk Seo

Publications and source records attributed to Insuk Seo.

At least 19 recordsLinked to original sources
arXiv2025

Towards Robust Mathematical Reasoning

Finding the right north-star metrics is highly critical for advancing the mathematical reasoning capabilities of foundation models, especially given that existing evaluations are either too easy or only focus on getting correct short answers. To address these issues, we present IMO-Bench, a suite of advanced reasoning benchmarks, vetted by a panel of top specialists and that specifically targets the level of the International Mathematical Olympiad (IMO), the most prestigious venue for young mathematicians. IMO-AnswerBench first tests models on 400 diverse Olympiad problems with verifiable short answers. IMO-Proof Bench is the next-level evaluation for proof-writing capabilities, which includes both basic and advanced IMO level problems as well as detailed grading guidelines to facilitate automatic grading. These benchmarks played a crucial role in our historic achievement of the gold-level performance at IMO 2025 with Gemini Deep Think (Luong and Lockhart, 2025). Our model achieved 80.0% on IMO-AnswerBench and 65.7% on the advanced IMO-Proof Bench, surpassing the best non-Gemini models by large margins of 6.9% and 42.4% respectively. We also showed that autograders built with Gemini reasoning correlate well with human evaluations and construct IMO-GradingBench, with 1000 human gradings on proofs, to enable further progress in automatic evaluation of long-form answers. We hope that IMO-Bench will help the community towards advancing robust mathematical reasoning and release it at https://imobench.github.io/.

cs.CL↗
arXiv2025

Cutoff Phenomenon for Inhomogeneous Nonlinear Recombination in Arbitrary Finite Product Spaces

In this article, we prove the cutoff phenomenon for a general class of the discrete-time nonlinear recombination models. This system models the evolution of a probability measure on a finite product space $S^n$ representing the state of spins on $n$ sites. Although its stationary distribution has a product structure, and its evolution is Markovian, the dynamics of the model is nonlinear. Consequently, the estimation of the mixing time becomes a highly non-trivial task. The special case with two spins and homogeneous stationary measure was considered in Caputo, Labb\'e, and Lacoin [The Annals of Applied Probability 35:1164-1197, 2025], where the cutoff phenomenon for the mixing behavior has been verified. In this article, we extend this result to the general case with finite spins and inhomogeneous stationary measure by developing a novel algebraic representation for the density fluctuation of the system with respect to its stationary state.

math.PR↗
arXiv2025

Eyring-Kramers Law for the Underdamped Langevin Process

Consider the underdamped Langevin process $(q(t),p(t))_{t\geq0}$ in $\R^d\times\R^d$. We derive the low-temperature asymptotic of its mean-transition time between basins of attraction for a double-well potential. This asymptotic is called Eyring-Kramers law and often relies in the literature on Potential theory tools which are ill-defined for hypoelliptic processes like the underdamped Langevin process. In this work, we implement a novel approach which circumvents the use of these traditional methods.

math.PR↗
arXiv2025

Exit Time Analysis For Kesten's Stochastic Recurrence Equations

Kesten's stochastic recurrent equation is a classical subject of research in probability theory and its applications. Recently, it has garnered attention as a model for stochastic gradient descent with a quadratic objective function and the emergence of heavy-tailed dynamics in machine learning. This context calls for analysis of its asymptotic behavior under both negative and positive Lyapunov exponents. This paper studies the exit times of the Kesten's stochastic recurrence equation in both cases. Depending on the sign of Lyapunov exponent, the exit time scales either polynomially or logarithmically as the radius of the exit boundary increases.

math.PR↗
arXiv2024

Metastability and time scales for parabolic equations with drift 2: the general time scale

Consider the elliptic operator given by \[ \mathscr{L}_\epsilon f=b\cdot\nabla f+\epsilon\Delta f \] for some smooth vector field $b:\mathbb{R}^d\to\mathbb{R}^d$ and $\epsilon>0$, and the initial-valued problem on $\mathbb{R}^d$ \[ \left\{\begin{aligned}&\partial_t u_\epsilon=\mathscr{L}_\epsilon u_\epsilon,\\ &u_\epsilon(0,\,\cdot)=u_0(\cdot), \end{aligned} \right. \] for some bounded continuous function $u_0$. Under the hypothesis that the diffusion on $\mathbb{R}^d$ induced by $\mathscr{L}_\epsilon$ has a Gibbs invariant measure of the form $\exp \{-U(x)/\epsilon\}dx$ for some smooth Morse potential function $U$, we provide the complete characterization of the multi-scale behavior of the solution $u_\epsilon$ in the regime $\epsilon\to0$. More precisely, we find the critical time scales $1\ll \theta_\epsilon^{(1)}\ll\cdots\ll \theta_\epsilon^{(q)}$ as $\epsilon\to0$, and the kernels $R_t^{(p)}:M_0\times M_0\to\mathbb{R}_+$, where $M_0$ denotes the set of local minima of $U$, such that \[ \lim_{\epsilon\to0}u_\epsilon(t\theta_\epsilon^{(p)},\,x)=\sum_{m'\in M_0}R_t^{(p)}(m,\,m')u_0(m'), \] for all $t>0$ and $x$ in the domain of attraction of $m$ for the dynamical system $\dot{x}(t)=b(x(t))$. We then complete the characterization of the solution $u_\epsilon$ by computing the exact asymptotic limit of the solution between time scales $\theta_\epsilon^{(p)}$ and $\theta_\epsilon^{(p+1)}$ for each $p$, where $\theta_\epsilon^{(0)}=1$ and $\theta_\epsilon^{(q+1)}=\infty$. Our analysis makes essential use of the hierarchical tree structure underlying the metastable behavior in different time-scales of the diffusion induced by $\mathscr{L}_\epsilon$. This result can be regarded as the precise refinement of Freidlin-Wentzell theory which was not known for more than a half century.

math.PR↗
arXiv2023

Asymptotic stability and cut-off phenomenon for the underdamped Langevin dynamics

In this article, we provide detailed analysis of the long-time behavior of the underdamped Langevin dynamics. We first provide a necessary condition guaranteeing that the zero-noise dynamical system converges to its unique attractor. We also observed that this condition is sharp for a large class of linear models. We then prove the so-called cut-off phenomenon in the small-noise regime under this condition. This result provides the precise asymptotics of the mixing time of the process and of the distance between the distribution of the process and its stationary measure. The main difficulty of this work relies on the degeneracy of its infinitesimal generator which is not elliptic, thus requiring a new set of methods.

math.PR↗
arXiv2023

Metastability and time scales for parabolic equations with drift 1: the first time scale

Consider the elliptic operator given by $$ \mathscr{L}_{\epsilon}f= {b} \cdot \nabla f + \epsilon \Delta f $$ for some smooth vector field $ b\colon \mathbb R^d \to\mathbb R^d$ and a small parameter $\epsilon>0$. Consider the initial-valued problem $$ \left\{ \begin{aligned} &\partial_ t u_\epsilon = \mathscr L_\epsilon u_\epsilon,\\ &u_\epsilon (0, \cdot) = u_0(\cdot), \end{aligned} \right. $$ for some bounded continuous function $u_0$. Denote by $\mathcal M_0$ the set of critical points of $b$ which are stable stationary points for the ODE $\dot x (t) = b (x(t))$. Under the hypothesis that $\mathcal M_0$ is finite and $ b = -(\nabla U + \ell)$, where $ \ell$ is a divergence-free field orthogonal to $\nabla U$, the main result of this article states that there exist a time-scale $\theta^{(1)}_\epsilon$, $\theta^{(1)}_\epsilon \to \infty$ as $\epsilon \rightarrow 0$, and a Markov semigroup $\{p_t : t\ge 0\}$ defined on $\mathcal M_0$ such that $$ \lim_{\epsilon\to 0} u_\epsilon (t\theta^{(1)}_\epsilon, x) =\sum_{m'\in \mathcal M_0} p_t(m, m')\, u_0( m'), $$ for all $t>0$ and $ x$ in the domain of attraction of $m$ for the ODE $\dot{x}(t)= b( x(t))$. The time scale $\theta^{(1)}$ is critical in the sense that, for all time scale $\varrho_\epsilon$ such that $\varrho_\epsilon \to \infty$, $\varrho_\epsilon/\theta^{(1)}_\epsilon \to 0$, $$ \lim_{\epsilon\to 0} u_\epsilon (\varrho_\epsilon, x)=u_0(m) $$ for all $x \in \mathcal D(m)$. Namely, $\theta_\epsilon^{(1)}$ is the first scale at which the solution to the initial-valued problem starts to change. In a companion paper [Landim, Lee, Seo, forthcoming] we extend this result finding all critical time-scales at which the solution $u_\epsilon$ evolves smoothly in time and we show that the solution $u_\epsilon$ is expressed in terms of the semigroup of some Markov chain taking values in sets formed by unions of critical points of $b$.

math.PR↗
arXiv2022

Approximation method to metastability: an application to non-reversible, two-dimensional Ising and Potts models without external fields

The main contribution of the current study is two-fold. First, we investigate the energy landscape of the Ising and Potts models on finite two-dimensional lattices without external fields in the low temperature regime. The complete analysis of the energy landscape of these models was unknown because of its complicated plateau saddle structure between the ground states. We characterize this structure completely in terms of a random walk on the set of sub-trees of a ladder graph. Second, we provide a considerable simplification of the well-known potential-theoretic approach to metastability. In particular, by replacing the role of variational principles such as the Dirichlet and Thomson principles with an $H^1$-approximation of the equilibrium potential, we develop a new method that can be applied to non-reversible dynamics as well in a simple manner. As an application of this method, we analyze metastable behavior of not only the reversible Metropolis-Hastings dynamics, but also of several interesting non-reversible dynamics associated with the low-temperature Ising and Potts models explained above, and derive the Eyring-Kramers law and the Markov chain model reduction of these models.

math.PR↗
arXiv2021

Non-reversible Metastable Diffusions with Gibbs Invariant Measure I: Eyring-Kramers Formula

In this article, we prove the Eyring-Kramers formula for non-reversible metastable diffusion processes that have a Gibbs invariant measure. Our result indicates that non-reversible processes exhibit faster metastable transitions between neighborhoods of local minima, compared to the reversible process considered in [Bovier, Eckhoff, Gayrard, and Klein, J. Eur. Math. Soc. 6: 399-424, 2004]. Therefore, by adding non-reversibility to the model, we can indeed accelerate the metastable transition. Our proof is based on the potential theoretic approach to metastability through accurate estimation of the capacity between metastable valleys. We carry out this estimation by developing a novel method to compute the sharp asymptotics of the capacity without relying on variational principles such as the Dirichlet principle or the Thomson principle.

math.PR↗
arXiv2021

Metastability of Ising and Potts models without external fields in large volumes at low temperatures

In this article, we investigate the energy landscape and metastable behavior of the Ising and Potts models on two-dimensional square or hexagonal lattices in the low temperature regime, especially in the absence of an external magnetic field. The energy landscape of these models without an external field is known to have a huge and complex saddle structure between ground states. In the small volume regime where the lattice is finite and fixed, the aforementioned complicated saddle structure has been successfully analyzed in [20] for two or three dimensional square lattices when the inverse temperature tends to infinity. In this article, we consider the large volume regime where the size of the lattice grows to infinity. We first establish an asymptotically sharp threshold such that the ground states are metastable if and only if the inverse temperature is larger than the threshold in a suitable sense. Then, we carry out a detailed analysis of the energy landscape and rigorously establish the Eyring-Kramers formula when the inverse temperature is sufficiently larger than the previously mentioned sharp threshold. The proof relies on detailed characterization of dead-ends appearing in the vicinity of optimal transitions between ground states and on combinatorial estimation of the number of configurations lying on a certain energy level.

math.PR↗
arXiv2021

Scaling Limit of Small Random Perturbation of Dynamical Systems

In this article, we prove that a small random perturbation of dynamical system with multiple stable equilibria converges to a Markov chain whose states are neighborhoods of the deepest stable equilibria, under a suitable time-rescaling, provided that the perturbed dynamics is reversible in time. Such a result has been anticipated from 1970s, when the foundation of mathematical treatment for this problem has been established by Freidlin and Wentzell. We solve this long-standing problem by reducing the entire analysis to an investigation of the solution of an associated Poisson equation, and furthermore provide a method to carry out this analysis by using well-known test functions in a novel manner.

math.PR↗
arXiv2021

Energy Landscape and Metastability of Stochastic Ising and Potts Models on Three-dimensional Lattices Without External Fields

In this study, we investigate the energy landscape of the Ising and Potts models on fixed and finite but large three-dimensional (3D) lattices where no external field exists and quantitatively characterize the metastable behavior of the associated Glauber dynamics in the very low temperature regime. Such analyses for the models with non-zero external magnetic fields have been extensively performed over the past two decades; however, models without external fields remained uninvestigated. Recently, the corresponding investigation has been conducted for the two-dimensional (2D) model without an external field, and in this study, we further extend these successes to the 3D model, which has a far more complicated energy landscape than the 2D one. In particular, we provide a detailed description of the highly complex plateau structure of saddle configurations between ground states and then analyze the typical behavior of the Glauber dynamics thereon. Thus, we acheive a quantitatively precise analysis of metastability, including the Eyring-Kramers law, the Markov chain model reduction, and a full characterization of metastable transition paths.

math.PR↗
arXiv2021

Condensation and Metastable Behavior of Non-reversible Inclusion Processes

In this article, we perform quantitative analyses of metastable behavior of an interacting particle system known as the inclusion process. For inclusion processes, it is widely believed that the system nucleates the condensation of particles because of the attractive nature of the interaction mechanism. The metastable behavior of the inclusion processes corresponds to the movement of the condensate on a suitable time scale, and the computation of the corresponding time scale and the characterization of the scaling limit of the condensate motion are the main problems in the study of metastability of inclusion processes. Previously, these problems were solved for reversible inclusion processes in [Bianchi, Dommers, and Giardinà, Electronic Journal of Probability, 22: 1-34, 2017], and the main contribution of the present study is to extend this analysis to a wide class of non-reversible inclusion processes. Non-reversibility is a major obstacle to analyzing such models, mainly because there is no closed-form expression of the invariant measure for the general case, and our main achievement is to overcome this difficulty. In particular, our results demonstrate that the time scale and limiting process of non-reversible inclusion processes are quantitatively and qualitatively different from those of reversible ones, respectively. We emphasize that, to the best of our knowledge, these results are the first rigorous quantitative results in the study of metastability when the invariant measure is not explicitly known. In addition, we consider the thermodynamic limit of metastable behavior of inclusion processes on large torus as in the paper [Armendáriz, Grosskinsky, and Loulakis, Probability Theory and Related Fields, 169: 105-175, 2017]. For this model, we observe three different time scales according to the level of asymmetry of the model.

math.PR↗
arXiv2020

Information Percolation and Cutoff for the Random-Cluster Model

We consider the Random-Cluster model on $(\mathbb{Z}/n\mathbb{Z})^d$ with parameters $p \in (0,1)$ and $q\ge 1$. This is a generalization of the standard bond percolation (with open probability $p$) which is biased by a factor $q$ raised to the number of connected components. We study the well known FK-dynamics on this model where the update at an edge depends on the global geometry of the system unlike the Glauber Heat Bath dynamics for spin systems, and prove that for all small enough $p$ (depending on the dimension) and any $q>1$, the FK-dynamics exhibits the cutoff phenomenon at $λ_{\infty}^{-1}\log n$ with a window size $O(\log\log n)$, where $λ_{\infty}$ is the large $n$ limit of the spectral gap of the process. Our proof extends the Information Percolation framework of Lubetzky and Sly [21] to the Random-Cluster model and also relies on the arguments of Blanca and Sinclair [4] who proved a sharp $O(\log n)$ mixing time bound for the planar version. A key aspect of our proof is the analysis of the effect of a sequence of dependent (across time) Bernoulli percolations extracted from the graphical construction of the dynamics, on how information propagates.

math.PR↗
arXiv2020

Non-reversible Metastable Diffusions with Gibbs Invariant Measure II: Markov Chain Convergence

This article considers a class of metastable non-reversible diffusion processes whose invariant measure is a Gibbs measure associated with a Morse potential. In a companion paper [32], we proved the Eyring-Kramers formula for the corresponding class of metastable diffusion processes. In this article, we further develop this result by proving that a suitably time-rescaled metastable diffusion process converges to a Markov chain on the deepest metastable valleys. This article is also an extension of [45], which considered the same problem for metastable reversible diffusion processes. Our proof is based on the recently developed resolvent approach to metastability.

math.PR↗
arXiv2020

Metastable behavior of weakly mixing Markov chains: the case of reversible, critical zero-range processes

We present a general method to derive the metastable behavior of weakly mixing Markov chains. This approach is based on properties of the resolvent equations and can be applied to metastable dynamics which do not satisfy the mixing conditions required in Beltr\'an and Landim (2010,2012) or in Landim et. al. (2020). As an application, we study the metastable behavior of critical zero-range processes. Let $r: S\times S\to \bb R_+$ be the jump rates of an irreducible random walk on a finite set $S$, reversible with respect to the uniform measure. For $\alpha >0$, let $g: \bb N\to \bb R_+$ be given by $g(0)=0$, $g(1)=1$, $g(k) = [k/(k-1)]^\alpha$, $k\ge 2$. Consider a zero-range process on $S$ in which a particle jumps from a site $x$, occupied by $k$ particles, to a site $y$ at rate $g(k) r(x,y)$. For $\alpha \ge 1$, in the stationary state, as the total number of particles, represented by $N$, tends to infinity, all particles but a negligible number accumulate at one single site. This phenomenon is called condensation. Since condensation occurs if and only if $\alpha\ge 1$, we call the case $\alpha =1$ critical. By applying the general method established in the first part of the article to the critical case, we show that the site which concentrates almost all particles evolves in the time-scale $N^2 \log N$ as a random walk on $S$ whose transition rates are proportional to the capacities of the underlying random walk.

math.PR↗
arXiv2019

Analysis of metastable behavior via solutions of Poisson equations

We herein review the recent progress on the study of metastability based on the analysis of solutions of Poisson equations related to the generators of the underlying metastable dynamics. This review paper is based on the joint work with Claudio Landim [24] and Fraydoun Rezakhanlou [26].

math.PR↗