#### Vadim Alexandrovich Malyshev

Russian scientist in the field of mathematics and mathematical physics, professor at Moscow State University, laureate of the State Prize of the RSFSR (1991).

MATHEMATICS, PHYSICS, LIFE

Conference dedicated to the 85-th Anniversary of

VADIM ALEXANDROVICH MALYSHEV(1938-2022)

June 26-30, 2023

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We study finite particle systems on the one-dimensional integer lattice, where each particle performs a continuous-time nearest-neighbour random walk, with jump rates intrinsic to each particle, subject to an exclusion interaction which suppresses jumps that would lead to more than one particle occupying any site. We show that the particle jump rates determine explicitly a unique partition of the system into maximal stable sub-systems, and that this partition can be obtained by a linear-time algorithm of elementary steps, as well as by solving a finite non-linear system. The internal configuration of each stable sub-system possesses an explicit product-geometric limiting distribution, and each stable sub-system obeys a strong law of large numbers with an explicit speed; the characteristic parameters of each stable sub-system are simple functions of the rate parameters for the corresponding particles. For the case where the entire system is stable, we prove a central limit theorem describing the fluctuations around the law of large numbers. Our approach draws on ramifications, in the exclusion context, of classical work of Goodman and Massey on unstable Jackson queueing networks.

In this talk we consider a model that can be interpreted either as a system of interacting spins, or as an interacting urn model. A particular interpretation depends on a motivating application and the model parameters. The model is technically flexible for modelling various types of collective behaviour (e.g. for modelling competition and cooperation, or a mixture of these). In the continuous time setting the model is stated in terms of a Markov chain with intensities given by state-dependent log-linear functions. We are interested in the long term behaviour of the model under various asymptotic regimes. It turns out that in many special cases a detailed description of the limit behaviour of the model is possible, which is illustrated by examples.

Cluster estimations of the moments or cumulants of the Gibbs fields (V. Malyshev, R. Minlos) have fundamental applications in population dynamics, demography and astrophysics (percolation phase transitions, dynamo problems etc.). The talk will contain a review of the old and new results in this area.

The sharp estimates of convergence rate in the classical Rényi theorem for random sums of random variables are stablished by means of the Stein techniques development. Special attention is paid to the model of exchangeable random variables. The integral probability metrics and the integration involving signed measures are employed as well.

We consider a random walk in an infinite strip of finite height. The transition probabilities satisfy the condition of spatial homogeneity with respect to one of the coordinates. We investigate the hitting probabilities of a finite set and the distribution of the first hitting point when the starting point is far from the reaching set.

Approximating a high-dimensional lattice by a regular tree ("Bethe lattice") is a classical device in Math Physics. But how do we best approximate the local geometry of a random cloud of points in space? The answer is the Poisson-weighted infinite tree (PWIT), introduced by Aldous and Steel for a completely different purpose.

We exhibit the effectiveness of this approximation by analyzing the greedy matching of two independent Poisson processes in Euclidean space under an asymmetric color restriction. Blue points can only match to red points, while red points can match to points of either color. It is unknown if there exist intensities for the red and blue processes for which all points are matched. We prove that for any fixed intensities, there are unmatched blue points in sufficiently high dimension. Our proof uses greedy matching on the PWIT, which can be analyzed via differential equations.

(Joint work with Alexander Holroyd and James Martin.)

I will explain the appearance of the KPZ scaling and Tracy-Widom distribution in the 2D and 3D Ising models. Based on a joint paper with Patrik Ferrari: The Airy2 process and the 3D Ising model, Journal of Physics A: Mathematical and Theoretical, 2023

We will discuss the topological expansion in unitary ensembles of random matrices and its relations to enumeration of graphs on Riemann surfaces. We will show how it is related to phase transitions and critical phenomena in random matrices

We consider a system of particles on a finite interval with Coulomb 3-dimensional interactions between close neighbours, i.e. only a few other neighbours apart. This model was introduced by Malyshev [Probl. Inf. Transm. 51 (2015) 31-36] to study the flow of charged particles. Notably even the nearest-neighbours interactions case, the only one studied previously, was proved to exhibit multiple phase transitions depending on the strength of the external force when the number of particles goes to infinity. Here we include as well interactions beyond the nearest-neighbours ones. Surprisingly but this leads to qualitatively new features even when the external force is zero. The order of the covariances of distances between pairs of consecutive charges is changed when compared with the former nearest-neighbours case, and moreover the covariances exhibit periodicity in sign: the interspacings are positively correlated if the number of interspacings between them is odd, otherwise, they are negatively correlated.

In the course of the proof we derive Gaussian approximation for the limit distribution for dependent variables described by a Gibbs distribution.

"In this talk, we discuss the evolution of the concept of the Gibbs measure, starting from its definition in a finite volume to the definition of a Gibbs random field that does not use the notion of potential. We also touch upon the question of non-Gibbsian random fields and Dobrushin's program on their Gibbsianess (see, for example, [4]).

All known approaches to defining a Gibbs random field were based on the notion of the Hamiltonian as the sum of interaction potentials. From the physical point of view, such a definition is very natural since it allows one to construct models of statistical physics with given properties. Meanwhile, from the mathematical point of view, it is necessary to have a definition of the Gibbs random field in terms of its intrinsic properties without the notion of the potential. Such a definition was given in the works [2, 3, 5], which made it possible to present the theory of Gibbs random fields in a purely probabilistic manner. The presence of a probabilistic definition allows investigating general questions for Gibbs random fields such as uniqueness, mixing properties and validity of limit theorems.

Note that, along the way, three important problems were solved: Dobrushin's problem on the description of random fields by systems of consistent one-point conditional distributions (see [1]), Ruelle's problem on the Gibbsian representation of specification (see [3]) and the problem of the description of a finite random field by its local characteristics (see [5]).

The talk is based on the joint works with Boris S. Nahapetian (Institute of Mathematics, NAS RA).

References

[1] S. Dachian and B.S. Nahapetian, Description of specifications by means of probability distributions in small volumes under condition of very week positivity. J. Stat. Phys. 117, 2004, 281-300

[2] S. Dachian and B.S. Nahapetian, On Gibbsiannes of Random Fields. Markov Process. Relat. Fields 15, 2009, 81-104

[3] S. Dachian and B.S. Nahapetian, On the relationship of energy and probability in models of classical statistical physics. Markov Process. Relat. Fields 25, 2019, 649-681

[4] R.L. Dobrushin and S.B. Shlosman, Gibbsian description of “non-Gibbsian” fields. Uspekhi Mat. Nauk 52, 1997, 45-58

[5] L. Khachatryan and B.S. Nahapetian, On the characterization of a finite random field by conditional distribution and its Gibbs form. J. Theor. Probab., 2019, 19 pp."

The testing the hypothesis about the belonging of the distribution of the observed data to a parametric gamma family of distribution functions is considered. The value of the vector parameter of the distribution of observations is supposed to be unknown. There, the Cramer-von- Mises (omega-square) statistic is investigated. The limiting Gaussian processes for empirical process depends on only one parameter. The tables of quantiles are presented for limit distribution of the Cramer- von-Mises statistic.

Very short arithmetical sums of periodic functions

Risk is present whenever the outcome is uncertain, whether favorable or unfavorable. Methods for transferring or distributing risk were practiced as long ago as the 3rd millennium BC, but risk theory (or actuarial science) emerged only in the 17th century. It has an interesting history consisting of 4 periods (deterministic, stochastic, financial, modern). The applied probability research domains such as insurance and finance, inventory and dams, queueing theory, reliability, population dynamics and some others can be considered as special cases of decision making under uncertainty (or risk management). The modern period is characterized by investigation of complex systems, application of intricate mathematical methods and emergence of new research directions.

We propose a definition of catastrophes and present our results on large deviations for Poisson processes with catastrophes that satisfy this definition. Our earlier work focused on (almost) uniformly distributed catastrophes, but the current paper extends the results to a larger class of catastrophes. We show that, regardless of the distribution of catastrophic events, the rate function remains the same. Additionally, we extend and generalize our previous results on the limiting behavior of the supremum of the considered processes. This is the joint work with Dr. Artem Logachov (Novosibirsk State University).

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A metric graph is a one-dimensional cellular complex. Each edge is a segment of a regular curve. Consider a random walk starting at some internal vertex. The times of passage of the edges are fixed, and a random walker can choose one of the incident edges at each vertex. U-turns at the inner points of the edges are prohibited. We consider a fixed real moment of time and count how many end positions of such a random walk are possible on the edges by this moment. The leading term of the asymptotics of this characteristic will be described as time increases for a certain class of directed graphs.

"We present the ingredients for an extension of the Third Law of Thermodynamics (Nernst heat postulate) to nonequilibrium processes. The central quantity is the excess heat which measures the quasistatic addition to the steady dissipative power when a parameter in the dynamics is changed slowly. We give conditions to prove for a class of driven Markov jump

processes that it vanishes at zero environment temperature. Furthermore, the nonequilibrium heat capacity goes to zero with temperature as well. Joint work with Faezeh Khodabandehlou (Leuven) and Karel Netocy (Prague).

arXiv:2210.09858v2 <https://arxiv.org/abs/2210.09858v2>"

We prove the orbital stability of solitons for the 2D Maxwell–Lorentz equations with extended charged spinning particle. The solitons are solutions corresponding to the uniform motion and rotation of the particle.

We prove the stability of solitons of the Maxwell--Lorentz equations with extended charged rotating particle. The solitons are stationary solutions which correspond to a uniform rotation of the particle. The charge density of the particle is spherically symmetric, and in this case the solitons exist for any angular velocity of rotation $\om\in\R^3$.

Our main result is that all the solitons are stable under two conditions on the charge density of the particle: the density is spherically symmetric and is not identically zero. To prove the stability, we construct the Hamilton-Poisson representation of the Maxwell-Lorentz system. The construction relies on the Hamilton least action principle and the Lie-Poincar\'e calculus. The constructed structure is degenerate and admits the Casimir invariants. This structure allows us to construct the Lyapunov function corresponding to a soliton. The function is a linear combination of the Hamiltonian with a suitable Casimir invariant. The function is conserved, and the soliton is its critical point. The key point of the proof is a lower bound for the Lyapunov function. This bound implies that the soliton is a strict local minimizer of the function. The proof of the bound for $\om=0$ is rather simple and use only the spherical symmetry. For $\om\ne 0$, the lower bound holds if and only if, additionally, the charge density is not identically zero. The proof relies on suitable spectral arguments including the Heinz inequality from the theory of interpolation.

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Usually the Markovian dynamics of an open quantum system arises in the Bogolubov-van Hove limit, which assumes the small coupling between the open system and its environment and appropriately long time-scale. For some specific physical models, namely, generalized spin-boson models in the rotating wave approximation, we show that under some natural conditions on so-called reservoir correlation functions the similar thing is true in all the orders of Bogolubov-van Hove perturbation theory. Markovian dynamics in the quantum case is typically defined as validity of the Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) master equation for the density matrix and regression formulae for multi-time correlations functions. We show that the GKSL equation is valid in all the orders of Bogolubov-van Hove perturbation theory. But to reproduce the right asymptotic precision at long times, one should use an initial condition different from the one for exact dynamics. Moreover, we show that the initial condition for this master equation even fails to be physical. It particular, it may have negative probabilities of level occupation. The regression formulae are not valid exactly, but should be renormalized in the similar manner as the initial condition. We call such a dynamics long-time Markovian one, because strictly speaking it is not Markovian, but all the non-Markovianity could be taken into account by such renormalization due-to non-Markovian dynamics in an initial layer, which is small with respect to the long time-scale separated by Bogolubov-van Hove scaling. This work is supported by the Russian Science Foundation under grant 19-11-00320.

Most results on Anderson localization in disordered quantum systems concern the so-called semiuniform localization of eigenfunctions, while the examples of models where the uniform localization occurs and is proved are quite rare. A class of such models with deterministic (e.g., quasi-periodic) potential will be presented; in these models, one can also establish an analog of the Minami-type estimates for the eigenvalues of the localized eigenfunctions.

We consider a harmonic crystal influenced by a thermal shock. We are interested in the kinetic temperature of the crystal on a long time scale. On the time scale of order N (N is a number of particles in the system), there is a thermal echo phenomenon which is related to the fluctuations of the temperature from an equilibrium. We will discuss this effect in detail. The recurrence of the temperature to the initial state is observed on the time scale exponentially big with N. We give lower and upper bounds on the mean recurrence time and discuss how this problem connected with the local principle of large deviations.

We consider two models: one describes the particle movement under the inﬂuence of external force and friction, and another one describes the movement of a particle, which is acted upon by the same external force but it additionally collides with other particles of much lighter masses. We establish conditions for these two models to be equivalent in some sense. We also considered deterministic and stochastic models for collisions, in ﬁrst case assuming that the time intervals between the collisions are constant, and in another case when these intervals are random independent random variables. For various examples of the external force we ﬁnd parameters which yield asymptotic equivalence of the velocities of the particle in diﬀerent models. We also provide conditions when the trajectories of the particles in diﬀerent models are close to each other in the Chebyshev norm over a certain ﬁnite period of time. Our results conﬁrm that a linear dissipative force such as e.g., friction, can well be modelled by the collisions with external light particles if their masses and the time-intervals between the collisions satisfy certain condition. The latter is proved here to be universal for diﬀerent forms of the external force.

The talk was initiated by the publication of V.A. Malyshev ``Stochastic Growth Models without Classical Branching Processes'' in the journal ``Markov Processes and Related Fields’’ (2022). We consider continuous-time branching random walks, which considerably extend the application area of branching processes by introducing particle transport into models with birth and death of particles. Such processes on multidimensional lattices are called branching random walks, and the points of the lattice at which the birth and death of particles can occur are called branching sources. We focus on branching random walks on multidimensional lattices under various assumptions about the properties of the underlying random walk. Particular attention is paid to study of the conditions under which there is the growth of a particle number at each point of the lattice, or the formation of particle clusters. Within the framework of branching random walk models, the effects of population growth of individuals can be explained in various applied studies. The work was supported by the Russian Science Foundation (grant 23-11-00375) and was performed at the Steklov Mathematical Institute of Russian Academy of Sciences.

We consider a model of branching random walk on an integer lattice Z^d with periodic sources of branching and death. We assume that the branching regime is supercritical and for a jump of random walk the Cramér condition is satisfied. Our main theorem describes the rate of propagation of front of particles population in branching random walk with periodic branching sources when time goes to infinity. The proof is based on fundamental results established for general branching random walks.

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Contact models in critical regime describe the populations of individuals belonging to different quasispecies in the case when there is a formal equilibrium between birth and death. We analyze the problem whether there is an invariant measure on configuration space corresponding to this formal equilibrium.

Рассматривается ветвящееся случайное блуждание (ВСБ) на одномерной целочисленной решетке Z c непрерывным временем. В основе ВСБ лежит простое симметричное случайное блуждание. В нуле возможно только деление частицы надвое, вне нуля возможна только гибель (поглощение) с некоторой случайной интенсивностью, которая ограничена единицей. Цель работы — исследовать условия возникновения экспоненциального роста количества частиц как в каждой точке Z, так и на всей решетке Z в описанной модели ВСБ. Для этого исследуются условия возникновения единственного положительного собственного значения эволюционного случайного оператора средних численностей частиц. В силу случайности поглощающей среды положительное собственное значение является случайной величиной. В работе исследуется как условия гарантирующие существование положительного собственного значения почти наверное, так и условия, приводящие к его существованию с некоторой вероятностью. Для оценки точности полученных условий проведено численное моделирование на языке программирования R.

Работа выполнена при поддержке РНФ, проект 23-11-00375 в Математическом институте им. В. А. Стеклова Российской академии наук.

Measure preserving dynamics could be associated canonically with the dynamic of metrics in the same space, which in its turn generated the new characteristics of the join system. For example, metric (or mm)-entropy of a space with measure and metric. It happened that asymptotics (w.r.t time) of those characteristics leave the dependence of initial metrics and become nontrivial invariants of the ergodic system. I had called such invariants --- catalytic invariants. The first example of it is so-called ""scaling entropy"" which was initiated by the author and developed during the last decade by author's school/ This notion drastically generates Shannon-Kolmogorov entropy and had series applications.

See recent survey by A.Vershik,.G.Veprev, P.Zatititsky. ""Dynamics of metrics and scaling entropy"". Russian Mathematical Surveys"" , 2023, №3, pp .53-114.

Эргодическую динамику с инвариантной мерой можно канонически ассоциировать с динамикой метрик в том же пространстве, которая в свою очередь порождает новые характеристики объединенной системы (например, её метрическую энтропию). Асимптотика (по времени) эти характеристик перестают зависеть от метрики и они становятся нетривиальными эргодическими инвариантами, которые я называю каталитическими. Их первым примером является понятие масштабированной энтропии, инициированное автором в последние годы. и существенно обобщающее энтропию Шеннона-Колмогорова. Первый обзор результатов на эту тему--- А.М.Вершик, Г.А.Вепрев, П.Б.Затицкий. см. ""Динамика метрик и масштабированная энтропия"", ""Успехах мат. наук"", 2023, №3, с.53-114.

Here we are looking at a system of an infinite number of particles. where no single particle can escape to infinity. We are going to define what it means energy escaping to infinity and apply this notion to the case where a force gives energy to one of the particles.

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