A5059713105- Stochastic processes and statistical mechanics
- Theoretical and Computational Physics
- Markov Chains and Monte Carlo Methods
- Advanced Mathematical Modeling in Engineering
- Advanced Thermodynamics and Statistical Mechanics
- Random Matrices and Applications
- Mathematical Dynamics and Fractals
- Spectral Theory in Mathematical Physics
- Quantum many-body systems
- Gamma-ray bursts and supernovae
- Astronomical Observations and Instrumentation
- nanoparticles nucleation surface interactions
- Geometry and complex manifolds
- Algorithms and Data Compression
- DNA and Biological Computing
- Stochastic processes and financial applications
- Diffusion and Search Dynamics
- Numerical methods in inverse problems
- Nonlinear Partial Differential Equations
- Computational Physics and Python Applications
- Advanced Condensed Matter Physics
- Physics of Superconductivity and Magnetism
- advanced mathematical theories
- Advanced Data Storage Technologies
- semigroups and automata theory
University of California, Los Angeles
2011-2024
Charles University
1999-2019
Wrocław University of Science and Technology
2019
AGH University of Krakow
2019
Center for Theoretical Physics
2018
Sewanee: The University of the South
2012
University of South Bohemia in České Budějovice
2009-2011
Technical University of Munich
2010
University of Warsaw
2005-2009
European Organization for Nuclear Research
2006-2009
Recent progress on the understanding of Random Conductance Model is reviewed and commented. A particular emphasis results scaling limit random walk among conductances for almost every realization environment, observations behavior effective resistance as well certain models gradient fields with non-convex interactions. The text an expanded version lecture notes a course delivered at 2011 Cornell Summer School Probability.
We consider the (unoriented) long-range percolation on ℤd in dimensions d≥1, where distinct sites x,y∈ℤd get connected with probability pxy∈[0,1]. Assuming pxy=|x−y|−s+o(1) as |x−y|→∞, s>0 and |⋅| is a norm distance ℤd, supposing that resulting random graph contains an infinite component C∞, we let D(x,y) be between x y measured C∞. Our main result that, for s∈(d,2d), D(x,y)=(log|x−y|)Δ+o(1), x,y∈C∞, Δ−1 binary logarithm of 2d/s o(1) quantity tending to zero |x−y|→∞. Besides its interest...
We consider the nearest-neighbor simple random walk on $Z^d$, $d\ge2$, driven by a field of i.i.d. conductances $\omega_{xy}\in[0,1]$. Apart from requirement that bonds with positive percolate, we pose no restriction law $\omega$'s. prove that, for a.e. realization environment, path distribution converges weakly to non-degenerate, isotropic Brownian motion. The quenched functional CLT holds despite fact local may fail in $d\ge5$ due anomalously slow decay probability returns starting point...
We consider the nearest-neighbor simple random walk on $\Z^d$, $d\ge2$, driven by a field of bounded conductances $\omega_{xy}\in[0,1]$. The conductance law is i.i.d. subject to condition that probability $\omega_{xy}>0$ exceeds threshold for bond percolation $\Z^d$. For environments in which origin connected infinity bonds with positive conductances, we study decay $2n$-step return $P_\omega^{2n}(0,0)$. prove $P_\omega^{2n}(0,0)$ constant times $n^{-d/2}$ $d=2,3$, while it $o(n^{-2})$...
These are expanded lecture notes for a minicourse taught at the "School on disordered media" Alfred Renyi institute in Budapest, January 2025.
We consider liquid-vapor systems in finite-volume V⊂d at parameter values corresponding to phase coexistence and study droplet formation due a fixed excess δN of particles above the ambient gas density. identify dimensionless Δ ∼ (δN)(d + 1)/d/V universal value Δc = Δc(d), show that dense occurs whenever > Δc, while, for < is entirely absorbed into gaseous background. When first forms, it comprises non-trivial, fraction particles. Similar reasoning applies generic two-phase including...
We consider the parabolic Anderson problem $\partial_t u = \kappa\Delta + \xi u$ on $(0,\infty) \times \mathbb{Z}^d$ with random i.i.d. potential $\xi (\xi(z))_{z\in \mathbb{Z}^d}$ and initial condition $u(0,\cdot) \equiv 1$. Our main assumption is that $\mathrm{esssup} \xi(0)=0$. Depending thickness of distribution $\mathrm{Prob} (\xi(0) \in \cdot)$ close to its essential supremum, we identify both asymptotics moments $u(t, 0)$ almostsure as $t \to \infty$ in terms variational problems. As...
We study the classical 120-degree and related orbital models. These are limits of quantum models which describe interactions among orbitals transition-metal compounds. demonstrate that at low temperatures these exhibit a long-range order arises via an "order by disorder" mechanism. This strongly indicates there is ordering in version models, notwithstanding recent rigorous results on absence spin systems.
We consider a quenched-disordered heteropolymer, consisting of hydrophobic and hydrophylic monomers, in the vicinity an oil-water interface. The heteropolymer is modeled by directed simple random walk$(i, S_i)_{i\epsilon\mathbb{N}}$ on $\mathbb{N} \times \mathbb{Z}$ with interaction given Hamiltonians $H_n^{\omega}(S) = \lambda \Sigma_{i=1}^n(\omega_i + h)\text{sign}(S_i)(n \epsilon \mathbb{N})$. Here, $\lambda$ h are parameters $(\omega_i)_{i\epsilon\mathbb{N}}$ i.i.d. $\pm1$-valued...
We present a general, rigorous theory of Lee-Yang zeros for models with first-order phase transitions that admit convergent contour expansions. derive formulas the positions and density zeros. In particular, we show that, without symmetry, curves on which lie are generically not circles, can have topologically nontrivial features, such as bifurcation. Our results illustrated in three complex field: low-temperature Ising Blume-Capel models, q-state Potts model large q.
We consider gradient fields (ϕx : x∈ℤd) whose law takes the Gibbs–Boltzmann form Z−1exp{−∑〈x, y〉V(ϕy−ϕx)}, where sum runs over nearest neighbors. assume that potential V admits representation V(η):=−log∫ϱ(d κ)exp[−½κη2], ϱ is a positive measure with compact support in (0, ∞). Hence, symmetric, but nonconvex general. While for strictly convex V's, translation-invariant, ergodic Gibbs measures are completely characterized by their tilt, as above may lead to several zero tilt. Still, every...
We study the asymptotic growth of diameter a graph obtained by adding sparse "long" edges to square box in $\Z^d$. focus on cases when an edge between $x$ and $y$ is added with probability decaying Euclidean distance as $|x-y|^{-s+o(1)}$ $|x-y|\to\infty$. For $s\in(d,2d)$ we show that for reduced side $L$ scales like $(\log L)^{\Delta+o(1)}$ where $\Delta^{-1}:=\log_2(2d/s)$. In particular, grows about fast typical two vertices at $L$. also ball radius $r$ intrinsic metric (infinite) will...
Nous considérons le champ libre gaussien discret (DGFF) dans des domaines DN⊆Z2 qu’on obtient, via une mise à l’échelle par N, partir de raisonnables D⊆R2. étudions les statistiques valeurs d’ordre logN en dessous du maximum absolu. Encodés un processus ponctuel sur D×R, la distribution spatiale ces ensembles niveaux proches D N et (en unités absolu) convergent, pour N→∞, loi vers produit gravité quantique critique Liouville (cLQG) ZD Rayleigh. La convergence est valable conjointement avec...
Nous considérons le champs Gaussien libre discret (DGFF) sur des versions renormalisées réseau carré de domaines continus suffisamment réguliers $D\subset\mathbb{C}$ et décrivons la limite d'échelle, incluant structure locale, lignes niveau lorsque que hauteur croît comme $\lambda$-fois du maximum absolu, pour tout $\lambda\in(0,1)$. montons que, dans position normalisée d'un point typique $x$ tiré aléatoirement cette ligne a loi mesure Gravité Quantique Liouville (LQG) $D$ avec paramètre...