Statistical physics of macromolecules
Lecture 1 Random walks on the lattice: Examples: 1D case – Dyck, Motzkin, Lukashevich. Recursion relations and their solutions. Necessary mathematical ingredients: Fourier and Laplace transforms, complex analysis (residues), Tauberian theorem, Dyson (self-consistency) equation vs recursion, conformal methods, scaling approach.
Lecture 2
Statistics of paths on lattices with constraints: Dirichlet and Neuman boundary conditions (image method), paths in a slit (ground state dominance), paths with an adsorbing center (paths on a half-line and in a 3D space), phase transition localization – delocalization. Computation of expectations from the partition function of paths’ counting and connection to statistics of uniform random walks on the lattice: mean displacement, return probability. Similarity and differences between counting statistics and probability (example: a uniform line vs a star).
Lectures 3-4
Paths’ statistics in arrays of randomly distributed traps (Balagurov-Vaks). Exact solution of 1D case and connection with Anderson localization in 1D random hopping model. Discrete Laplacians and simple random graphs. Spectral methods in statistics of paths. Spectral statistics of random interval model and its connection to modular forms.
Lectures 5-6
Paths in a continuous space. Fractal structure of trajectories: fractal dimension, Koch snowflake, notion about the Fractal Brownian motion. Path integrals for random walks, Lagrangian, principle of minimum of action, Euler equation. Quantum vs classical solutions. Examples from geometric optics and the trajectory of a growing cow on the field. Eiconal equation. 2D random walk with a fixed algebraic area. Structure (metrics) of space vs potential (circular geodesics for a cow and a Larmor orbit).
Lecture 7
Random walks from Langevin equation. Fokker-Plank vs Schrodinger equations. Random walks in potentials. Adsorption of a periodic copolymer on a line in 1D and on a point well in 3D. Random walks in an attractive harmonic potential – generation of compact paths with a fractal dimension larger than 2. Action with a nonlocal kernel (with a memory).
Lecture 8
Branching paths near boundaries. Critical exponents. Kinetic approach to random graphs and percolation.
Lectures 9-10
Polymers as random walks without self-intersections. Flory mean-field approach. Perturbation theory. Real space renormalization group and e-expansion. Relation to n-component scalar field theory in the limit n → 0.
Lecture 11
Methods of self-consistent field theory. Coil-globule phase transition. Order of a phase transition. Connection to the problem of random walk adsorption at the potential well.
Lecture 12
Random walks on manifolds. Beltrami-Laplace operators on Riemann manifolds: random walks in flat geometry, on sphere and on pseudosphere. Poincare model of the surface of constant negative curvature. Brownian bridge on the surface of constant negative curvature and extremal statistics. Watermelons (multiple bridges) in the space of a constant negative curvature
Lecture 2
Statistics of paths on lattices with constraints: Dirichlet and Neuman boundary conditions (image method), paths in a slit (ground state dominance), paths with an adsorbing center (paths on a half-line and in a 3D space), phase transition localization – delocalization. Computation of expectations from the partition function of paths’ counting and connection to statistics of uniform random walks on the lattice: mean displacement, return probability. Similarity and differences between counting statistics and probability (example: a uniform line vs a star).
Lectures 3-4
Paths’ statistics in arrays of randomly distributed traps (Balagurov-Vaks). Exact solution of 1D case and connection with Anderson localization in 1D random hopping model. Discrete Laplacians and simple random graphs. Spectral methods in statistics of paths. Spectral statistics of random interval model and its connection to modular forms.
Lectures 5-6
Paths in a continuous space. Fractal structure of trajectories: fractal dimension, Koch snowflake, notion about the Fractal Brownian motion. Path integrals for random walks, Lagrangian, principle of minimum of action, Euler equation. Quantum vs classical solutions. Examples from geometric optics and the trajectory of a growing cow on the field. Eiconal equation. 2D random walk with a fixed algebraic area. Structure (metrics) of space vs potential (circular geodesics for a cow and a Larmor orbit).
Lecture 7
Random walks from Langevin equation. Fokker-Plank vs Schrodinger equations. Random walks in potentials. Adsorption of a periodic copolymer on a line in 1D and on a point well in 3D. Random walks in an attractive harmonic potential – generation of compact paths with a fractal dimension larger than 2. Action with a nonlocal kernel (with a memory).
Lecture 8
Branching paths near boundaries. Critical exponents. Kinetic approach to random graphs and percolation.
Lectures 9-10
Polymers as random walks without self-intersections. Flory mean-field approach. Perturbation theory. Real space renormalization group and e-expansion. Relation to n-component scalar field theory in the limit n → 0.
Lecture 11
Methods of self-consistent field theory. Coil-globule phase transition. Order of a phase transition. Connection to the problem of random walk adsorption at the potential well.
Lecture 12
Random walks on manifolds. Beltrami-Laplace operators on Riemann manifolds: random walks in flat geometry, on sphere and on pseudosphere. Poincare model of the surface of constant negative curvature. Brownian bridge on the surface of constant negative curvature and extremal statistics. Watermelons (multiple bridges) in the space of a constant negative curvature
Lecturer
Serguei Nechaev
Date
27th February ~ 24th April, 2026
Location
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| Tuesday | 10:40 - 12:15 | Shuangqing-B627 | ZOOM 07 | 559 700 6085 | BIMSA |
| Friday | 17:05 - 18:40 | Shuangqing-B627 | ZOOM 07 | 559 700 6085 | BIMSA |
Video Public
Yes
Notes Public
Yes
Language
English
Lecturer Intro
I graduated in 1985 from the Department of Physics at Moscow State University. From 1991 to 2007, I was affiliated with the Landau Institute for Theoretical Physics.
Since 2008, I have been working in France, where I currently hold the position of Director of Research at CNRS (National Center for Scientific Research, France). I am based at the LPTMS (Laboratory of Theoretical Physics and Statistical Models), a laboratory affiliated with both CNRS and the University of Paris-Saclay (https://www.lptms.universite-paris-saclay.fr/).
From 2015 to 2022, I served as the Director of the Interdisciplinary Scientific Center Poncelet, an International Research Laboratory of CNRS, located at the Independent University of Moscow (Moscow, Russia).
I have authored more than 140 scientific publications in Web of Science journals. My current scientific interests are systematized below, where the list of main publications is presented. These publications constitute the core of references related to the course “Statistics and Topology of Random Paths”.