IS20: Non-Equilibrium Statistical Mechanics

Organizer: Makoto Katori (Chuo University)

Stochastic and dynamical approaches to non-Hermitian matrix-valued processes

Makoto Katori

One of the main topics of non-equilibrium statistical mechanics is to establish useful mathematical descriptions of open systems, in which flows of energy, particles, and information to external degrees of freedom play essential roles. For this purpose, a variety of models associated with non-Hermitian operators and matrices have been extensively studied in a new research field called non-Hermitian physics. Motivated by such a new trend, we will report the stochastic analysis of non-Hermitian matrix-valued Brownian motion (BM) and related stochastic processes. In contrast to Dyson’s BM models describing the eigenvalue processes of the Hermitian matrix-valued BMs, the stochastic differential equations (SDEs) of the eigenvalue processes are coupled with those of the eigenvector-overlap processes. The latter cause random time-change to the former. We also discuss non-normal and defective matrix-valued dynamical systems in order to demonstrate the importance of the notion of pseudospectra in addition to eigenvalues. This talk is based on the joint work with Syota Esaki (Oita), Satoshi Yabuoku (Fukuoka) \([1]\) and Jacek Małecki (Wrocław), and with Saori Morimoto (Chuo) and Tomoyuki Shirai (Kyushu) \([2,3]\).

Bibliography

\([1]\) Syota Esaki, Makoto Katori, Satoshi Yabuoku. "Eigenvalues, eigenvector-overlaps, and regularized Fuglede–Kadison determinant of the non-Hermitian matrix-valued Brownian motion." arXiv:math.PR/2306.00300

\([2]\) Saori Morimoto, Makoto Katori, Tomoyuki Shirai. "Eigenvalue and pseudospectrum processes generated by nonnormal Toeplitz matrices with rank 1 perturbations." arXiv:math-ph/2401.08129

\([3]\) Saori Morimoto, Makoto Katori, Tomoyuki Shirai. "Generalized eigenspaces and pseudospectra of nonnormal and defective matrix-valued dynamical systems." arXiv:math-ph/2411.06472

Collisions of the supercritical Keller-Segel particle system

Yoan Tardy

We study a particle system naturally associated to the \(2\)-dimensional Keller-Segel equation. It consists of \(N\) Brownian particles in the plane, interacting through a binary attraction in \(\theta /Nr\), where \(r\) stands for the distance between two particles. When the intensity \(\theta\) of this attraction is greater than \(2\), this particle system explodes in finite time. We assume that \(N>3\theta\) and study in details what happens near explosion. There are two slightly different scenarios, depending on the values of \(N\) and \(\theta\), here is one: at explosion, a cluster consisting of precisely \(k_0\) particles emerges, for some deterministic \(k_0\ge 7\) depending on \(N\) and \(\theta\). Just before explosion, there are infinitely many \((k_0-1)\)-ary collisions. There are also infinitely many \((k_0-2)\)-ary collisions before each \((k_0-1)\)-ary collision. And there are infinitely many binary collisions before each \((k_0-2)\)-ary collision. Finally, collisions of subsets of \(3,\dots,k_0-3\) particles never occur. The other scenario is similar except that there are no \((k_0-2)\)-ary collisions.

Theorem.(partially informal) Assume that \(\theta \geq 2\), that \(N>3\theta\) and define \(k_0 =\lceil 2N/\theta\rceil \in \{ 7,\dots N\}\), \(k_1 = k_0-1\) and \(k_2\in \{k_0-1,k_0-2\}\) depending on \(\theta\) and \(N\). Consider the \(KS(\theta ,N)\)-process \((X_t)_{t\ge 0}\) which is a solution in a weak sense of the SDE \[\mbox{ for all } i\in \{1,\dots, N\}, \quad {\rm d}X^i_t = {\rm d}B^i_t - \frac{\theta}{N} \sum_{j\ne i} \frac{X^i_t - X^j_t}{\|X^i_t - X^j_t\|^2} {\rm d}t.\] For all \(x \in E_2\), we \(\mathbb{P}_x\)-a.s. have the following properties:

\(i\) \(\zeta\) is finite and \(X_{\zeta- } =\lim _{t \rightarrow \zeta- } X_{t}\) exists for the usual topology of \((\mathbb{R}^2)^N\);

\(ii\) there is \(K_0\subset \{ 1,\dots, N \}\) with cardinal \(|K_0| = k_0\) such that there is a \(K_0\)-collision in the configuration \(X_{\zeta-}\), and for all \(K\subset \{ 1,\dots,N \}\) such that \(|K| >k_0\), there is no \(K\)-collision in the configuration \(X_{\zeta -}\);

\(iii\) for all \(t\in [0,\zeta)\) and all \(K \subset K_0\) with cardinal \(|K| = k_1\), during \((t,\zeta)\), there is an infinite number of \(K\)-collisions and none of these instants of \(K\)-collision is isolated;

\(iv\) if \(k_2= k_0-2\), then for all \(L\subset K\subset K_0\) such that \(|L| = k_2\) and \(|K| = k_1\), for all instant \(t \in (0,\zeta)\) of \(K\)-collision and all \(s\in [0,t)\), during \((s,t)\), there is an infinite number of \(L\)-collisions and none of these instants of \(L\)-collision is isolated;

\(v\) for all \(K\subset \{ 1,\dots ,N \}\) with cardinal \(|K| \in \{ 3,\dots, k_2-1 \}\), during \([0,\zeta )\), there is no \(K\)-collision ;

\(vi\) for all \(L\subset K\subset K_0\) such that \(|L| = 2\) and \(|K| = k_2\), for all instant \(t \in (0,\zeta)\) of \(K\)-collision and all \(s\in [0,t)\), during \((s,t)\), there is an infinite number of \(L\)-collisions and none of these instants of \(L\)-collision is isolated.

Bibliography

\([1]\) P. Cattiaux, L. Pédèches, The 2-D stochastic Keller-Segel particle model: existence and uniqueness, ALEA, Lat. Am. J. Probab. Math. Stat. 13 (2016), 447–463.

\([2]\) N. Fournier, B. Jourdain, Stochastic particle approximation of the Keller-Segel equation and two-dimensional generalization of Bessel processes, Ann. Appl. Probab. 27 (2017), 2807–2861.

\([3]\) N. Fournier, Y. Tardy. Collisions of the supercritical Keller-Segel particle system. To appear in J. Eur. Math. Soc. Available on ArXiv: 2110.08490 math.PR, 2021.

\([4]\) M. Fukushima, Y. Oshima, M. Takeda, Dirichlet forms and symmetric Markov processes. Second revised and extended edition. Walter de Gruyter, 2011.

\([5]\) E.F. Keller, L.A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol. 26 (1970), 399–415.

Assistant Professor

Sung-Soo Byun

Title Free energy expansions of non-Hermitian random matrix ensembles
Abstract In the theory of two-dimensional Coulomb gases, the asymptotic expansion of the free energy as the system size \(N\) grows is a central and longstanding problem. The coefficients in this expansion are believed to encode fundamental information about the model, including potential-theoretic quantities, topological invariants, and conformal geometric properties \[3,5\]. While the first three terms in the expansion are now well understood from a mathematical perspective \[4\], the full structure of the expansion remains largely open.

In this talk, I will discuss non-Hermitian random matrix ensembles belonging to the complex and symplectic Ginibre symmetry classes, which can be realised as determinantal and Pfaffian Coulomb gases in the complex plane. I will present recent progress on the free energy expansions of these models, and explain how these results connect to developments in integrable probability.

Bibliography

\([1]\) S.-S. Byun, N.-G. Kang and S.-M. Seo, Partition functions of determinantal and Pfaffian Coulomb gases with radially symmetric potentials, Comm. Math. Phys. 401 (2023), 1627–1663.

\([2]\) S.-S Byun, S.-M. Seo and M. Yang, Free energy expansions of a conditional GinUE and large deviations of the smallest eigenvalue of the LUE, arXiv:2402.18983.

\([3]\) B. Jancovici, G. Manificat and C. Pisani, Coulomb systems seen as critical systems: finite-size effects in two dimensions, J. Stat. Phys. 76 (1999), 307–329.

\([4]\) T. Leblé and S. Serfaty, Large deviation principle for empirical fields of log and Riesz gases, Invent. Math. 210 (2017), 645–757.

\([5]\) A. Zabrodin and P. Wiegmann, Large-N expansion for the 2D Dyson gas, J. Phys. A 39 (2006), 8933–8964.