Beijing Institute of Mathematical Sciences and Applications

14th February ~ 13th May, 2025

Lecture 1 Combinatorics of paths 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 methods.
Lecture 2 Combinatorics 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 difference 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 with modular functions. Notion about the Dedekind function and its basic properties.
Lecture 5 Laplacians on graphs. Kirchhoff matrix theorem. Laplacians on trees and Catalan numbers. Paths’ localization on uniform trees (tree with a heavy root).
Lectures 6 - 7 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. 2D random walk with a fixed algebraic area. Structure (metrics) of space vs potential.
Lecture 8 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 9 Branching paths near boundaries. Critical exponents. Kinetic approach to random graphs and percolation.
Lecture 10 Non-Gaussian scaling exponents of “stretched paths” near convex boundaries. Scaling approach and optimal fluctuation. Stretched paths as random walks with extremal gyration radius. Notion about Kardar-Parisi-Zhang equation and related models.
Lectures 11 - 12 Polymers as random walks without self-intersections. Flory mean-field approach. Perturbation theory. Real space renormalization group and $\epsilon$-expansion. Relation to n-component scalar field theory in the limit $n\rightarrow 0$.
Lecture 13 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 constant negative curvature.
Lectures 14 - 15 Random walks on a surface of a D-dimensional sphere: Hermite polynomials as eigenfunctions in the limit $D\gg 1$. Target space structure associated with Hermite polynomials. Random walks on a nonuniform tree and non-Gaussian statistics. Airy functions. Connection with 1D Dyck paths in a constant magnetic field (area-preserved random walks). Magnetic “q-walks”.
Lecture 16 Random walks on groups. Random walk on a free group as a random walk on a uniform tree. Random walk on a group as a growth of a two-row two-color pile. Growth of a multiple-row pile. One-color pile and a “physicist’s view” on RSK algorithm of longest increasing subsequence growth. Recursion relations and Fisher equation. Derivation of the mean height.
Lecture 17 Combinatorics of heaps. Normal ordering of words: “Mikado” algorithm. Spectral methods for transfer-matrix approach. Chebyshev polynomials. Connection with Lukashevich paths. Heaps of pieces as approximants of braids. Word counting in braid groups. Random walks on braid groups. Brownian bridge on braid group.
Lecture 18 Random walk with topological constraints. Winding around a single obstacle. Bohm-Aharonov effect. Conformal approach.
Lectures 19 - 20 Random walk on a doubly punctured plane and in the lattice of obstacles. Nonabelian nature of the underlying group. Pochhammer contour. Conformal approach and conformal invariance of the random walk. Random walk on the universal covering space and in the lattice of obstacles. Topological invariants. Scaling relation for an unentangled loop in the lattice of obstacles. Random walk on a 3-string braid group, 3-string braid group as central extension of $SL(2,\mathbb{Z})$. “Magnetic random walks” on $SL(2,\mathbb{Z})$.
Lectures 21 - 22 Lattice knots as disordered systems. Algebraic topological invariants: skein relations and Reidemeister moves. Kauffman bracket invariant as a partition function. Notion about statistical models: Ising and Potts models. Lattice knots and Kauffman invariant as a partition function of a disordered Potts model.
Lecture 23 Conditional probability of lattice knots. Brownian bridges in space of knots. Manifestation of Brownian bridges in genomics. Basic notion about the topological structure of chromosomes. Compact (“globular”) state of polymers. Crumpled globule vs ordinary globule.
Lecture 24 Golden and Silver rations in phyllotaxis problem. Optimal flow in a modular domain. Connection with tessellation of Euclidean plane by equilateral and isosceles triangles. Connection with contents of lectures 19 – 20.

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