### Published papers

1. Matthias Birkner, Jiří Černý, Andrej Depperschmidt: Random walks in dynamic random environments and ancestry under local population regulation.
Electronic Journal in Probability 21, No. 38, 1-43 (2016)
[ Abstract, pdf, arXiv, printed version ]
We consider random walks in dynamic random environments, with an environment generated by the time-reversal of a Markov process from the 'oriented percolation universality class'. If the influence of the random medium on the walk is small in space-time regions where the medium is 'typical', we obtain a law of large numbers and an averaged central limit theorem for the walk via a regeneration construction under suitable coarse-graining.
Such random walks occur naturally as the spatial embedding of an ancestral lineage in spatial population models with local regulation. We verify that our assumptions hold for logistic branching random walks when the population density is sufficiently high, thus partly settling a question from Depperschmidt (2008) on the behaviour of their ancestral lines.
2. Jiří Černý, Artem Sapozhnikov: Mixing time for the random walk on the range of the random walk on tori.
Elecronic Communications in Probability 21, No. 26, 1-10 (2016).
[ Abstract, pdf, arXiv, printed version ]
Consider the subgraph of the discrete $$d$$-dimensional torus of size length $$N$$, $$d\geq 3$$, induced by the range of the simple random walk on the torus run until the time $$uN^d$$. We prove that for all $$d\geq 3$$ and $$u>0$$, the mixing time for the random walk on this subgraph is of order $$N^2$$ with probability at least $$1 - Ce^{-(\log N)^2}$$.
3. Jiří Černý, Augusto Teixeira: Random walks on torus and random interlacements: macroscopic coupling and phase transition.
Annals of Applied Probability 26, No. 5, 2883–2914 (2016).
[ Abstract, pdf, arXiv, printed version ]
For $$d\ge 3$$ we construct a new coupling of the trace left by a random walk on a large $$d$$-dimensional discrete torus with the random interlacements on $$\mathbb{Z}^d$$. This coupling has the advantage of working up to \emph{macroscopic} subsets of the torus. As an application, we show a sharp phase transition for the diameter of the component of the vacant set on the torus containing a given point. The threshold where this phase transition takes place coincides with the critical value $$u_\star(d)$$ of random interlacements on $$\mathbb Z^d$$. Our main tool is a variant of the \emph{soft-local time} coupling technique of [PT12].
4. Jiří Černý, Tobias Wassmer: Aging of the Metropolis dynamics on the Random Energy Model.
Probability Theory and Related Fields 167, No. 1, 253-303 (2017).
[ Abstract, pdf, arXiv, printed version ]
We study the Metropolis dynamics of the simplest mean-field spin glass model, the Random Energy Model. We show that this dynamics exhibits aging by showing that the properly rescaled time change process between the Metropolis dynamics and a suitably chosen fast' Markov chain converges in distribution to a stable subordinator. The rescaling might depend on the realization of the environment, but we show that its exponential growth rate is deterministic.
5. Jiří Černý, Tobias Wassmer: Randomly trapped random walks on $$\mathbb Z^d$$.
Stochastic Processes and their Applications 125, No. 3, 1032-1057 (2015).
[ Abstract, pdf, arXiv ]
We give a complete classification of scaling limits of randomly trapped random walks and associated clock processes on $$\mathbb Z^d$$, $$d\ge 2$$. Namely, under the hypothesis that the discrete skeleton of the randomly trapped random walk has a slowly varying return probability, we show that the scaling limit of its clock process is either deterministic linearly growing or a stable subordinator. In the case when the discrete skeleton is a simple random walk on $$\mathbb Z^d$$, this implies that the scaling limit of the randomly trapped random walk is either Brownian motion or the Fractional Kinetics process, as conjectured in [BCCR13].
6. Gérard Ben Arous, Manuel Cabezas, Jiří Černý, Roman Royfman: Randomly Trapped Random Walks.
Annals of Probability 43, No. 5, 2405-2457 (2015).
[ Abstract, pdf, arXiv, printed version]
We introduce a general model of trapping for random walks on graphs. We give the possible scaling limits of these Randomly Trapped Random Walks on Z. These scaling limits include the well known Fractional Kinetics process, the Fontes-Isopi-Newman singular diffusion as well as a new broad class we call Spatially Subordinated Brownian Motions. We give sufficient conditions for convergence and illustrate these on two important examples.
7. Matthias Birkner, Jiří Černý, Andrej Depperschmidt, Nina Gantert: Directed random walk on an oriented percolation cluster.
Electronic Journal of Probability 18, No.80, 1-35 (2013).
[ Abstract, pdf, arXiv, printed version]
We consider directed random walk on the infinite percolation cluster generated by supercritical oriented percolation, or equivalently the space-time embedding of the ancestral lineage' of an individual in the stationary discrete-time contact process. We prove a law of large numbers and an annealed central limit theorem (i.e., averaged over the realisations of the cluster) with a regeneration approach. Via an analysis of joint renewals of two independent walks on the same cluster, we obtain furthermore a quenched central limit theorem (i.e. for almost any realisation of the cluster) in dimensions 1+d with d ≥ 2.
8. Jiří Černý, Augusto Teixeira: From random walk trajectories to random interlacements.
(Expanded lecture notes for the XV Brazilian School of Probability in 2011),
Ensaios Matemáticos 23 (2012).
[ Abstract, pdf ]
We review and comment recent research on random interlacements model introduced by A.-S. Sznitman in [Szn10]. A particular emphasis is put on motivating the definition of the model via natural questions concerning geometrical/percolative properties of random walk trajectories on finite graphs, as well as on presenting some important techniques used in random interlacements' literature in the most accessible way. This text is an expanded version of the lecture notes for the mini-course given at the XV Brazilian School of Probability in 2011.
9. Jiří Černý, Serguei Popov: On the internal distance in the interlacement set.
Electronic Journal of Probability 17, No.29, 1-25 (2012).
[ Abstract, pdf, printed version ]
We prove a shape theorem for the internal (graph) distance on the interlacement set Iu of the random interlacement model on Zd, d ≥ 3. We provide large deviation estimates for the internal distance of distant points in this set, and use these estimates to study the internal distance on the range of a simple random walk on a discrete torus.
10. Jiří Černý, Augusto Teixeira: Critical window for the vacant set left by random walk on random regular graphs.
Random Structures and Algorithms 43, No. 3, 313-337 (2013).
[ Abstract, pdf, printed version ]
We consider the simple random walk on a random d-regular graph with n vertices, and investigate percolative properties of the set of vertices not visited by the walk until time un, where u>0 is a fixed positive parameter. It was shown in [CTW11] that this so-called vacant set exhibits a phase transition at u=u*: there is a giant component if u<u* and only small components when u>u*. In this paper we show the existence of a critical window of size n-1/3 around u*. In this window the size of the largest cluster is of order~n2/3.
11. Antonio Auffinger, Gérard Ben Arous, Jiří Černý: Random matrices and complexity of spin glasses.
Communications on Pure and Applied Mathematics 66, No.2, 165-201 (2013)
[ Abstract, pdf, printed version ]
We give an asymptotic evaluation of the complexity of spherical p-spin spin-glass models via random matrix theory. This study enables us to obtain detailed information about the bottom of the energy landscape, including the absolute minimum (the ground state), the other local minima, and describe an interesting layered structure of the low critical values for the Hamiltonians of these models. We also show that our approach allows us to compute the related TAP-complexity and extend the results known in the physics literature. As an independent tool, we prove a LDP for the k-th largest eigenvalue of the GOE, extending the results of Ben Arous, Dembo and Guionnett (2001).
12. Jiří Černý: On two-dimensional random walk among heavy-tailed conductances.
Electronic Journal of Probability 16, 293-313 (2011).
[ Abstract, pdf, printed version ]
We consider a random walk among unbounded random conductances on the two-dimensional integer lattice. When the distribution of the conductances has an infinite expectation and a polynomial tail, we show that the scaling limit of this process is the fractional kinetics process. This extends the results of the paper [BC09] where a similar limit statement was proved in dimension d ≥ 3. To make this extension possible, we prove several estimates on the Green function of the process killed on exiting large balls.
13. Jiří Černý, Augusto Teixeira, David Windisch: Giant vacant component left by a random walk in a random d-regular graph.
Annales d'Institut Henri Poincare (B) Probability and Statistics 47, No. 4, 929-968 (2011).
[ Abstract, pdf, printed version ]
We study the trajectory of a simple random walk on a d-regular graph with d >2 and locally tree-like structure as the number n of vertices grows. Examples of such graphs include random d-regular graphs and large girth expanders. For these graphs, we investigate percolative properties of the set of vertices not visited by the walk until time un, where u>0 is a fixed positive parameter. We show that this so-called vacant set exhibits a phase transition in u in the following sense: there exists an explicitly computable threshold u* ∈ (0,∞) such that, with high probability as n grows, if u<u*, then the largest component of the vacant set has a volume of order n, and if u>u*, then it has a volume of order log n. The critical value u* coincides with the critical intensity of a random interlacement process on a d-regular tree. We also show that the random interlacements model describes the structure of the vacant set in local neighbourhoods.
14. Martin T. Barlow, Jiří Černý: Convergence to fractional kinetics for random walks associated with unbounded conductances.
Probability Theory and Related Fields 149, No. 3-4, 639-673 (2011).
[ Abstract, pdf, printed version ]
We consider a random walk among unbounded random conductances whose distribution has infinite expectation and polynomial tail. We prove, that the scaling limit of this process is a Fractional-Kinetics process -- that is the time change of a d-dimensional Brownian motion by the inverse of an independent α-stable subordinator. We further show, that the same process appears in the scaling limit of the non-symmetric Bouchaud's trap model.

Erratum to this paper [ pdf ]
15. Jiří Černý: Another view on aging in the REM.
in Spin Glasses: Statics and Dynamics Summer School, Paris 2007. Progress in Probability 62, 85-101, Birkhauser (2009)
[ Abstract, pdf, printed version ]
We give a new proof of aging for a version of a Glauber dynamics in the Random Energy Model. The proof uses ideas that were developed in [BBC08] for studying the dynamics of a p-spin Sherrington-Kirkpatrick spin glass.
16. Gérard Ben Arous, Anton Bovier, Jiří Černý: Universality of the REM for dynamics of mean-field spin glasses.
Communications in Mathematical Physics 61, No. 3, 289-329 (2008).
[ Abstract, pdf, printed version ]
We consider a version of a Glauber dynamics for a p-spin Sherrington-Kirkpatrick model of a spin glass that can be seen as a time change of simple random walk on the N-dimensional hypercube. We show that, for any p ≥ 3 and any inverse temperature β > 0, there exist constants ξ > 0, such that for all exponential time scales, exp(γN), with γξ, the properly rescaled clock process (time-change process), converges to an α-stable subordinator where α = γ/ β2< 1. Moreover, the dynamics exhibits aging at these time scales with time-time correlation function converging to the arcsine law of this α-stable subordinator. In other words, up to rescaling, on these time scales (that are shorter than the equilibration time of the system), the dynamics of p-spin models ages in the same way as the REM, and by extension Bouchaud's REM-like trap model, confirming the latter as a universal aging mechanism for a wide range of systems. The SK model (the case p = 2) seems to belong to a different universality class.
17. Gérard Ben Arous, Anton Bovier, Jiří Černý: Universality of random energy model like aging in mean field spin glasses.
Journal of Statistical Mechanics: Theory and Experiment 2008, No. 4, L04003 (2008).
[ Abstract, pdf, printed version ]
Aging has become the paradigm to describe dynamical behavior of glassy systems, and in particular spin glasses. Trap models have been introduced as simple caricatures of effective dynamics of such systems. In this Letter we show that in a wide class of mean field models and on a wide range of time scales, aging occurs precisely as predicted by the REM-like trap model of Bouchaud and Dean. This is the first rigorous result about aging in mean field models except for the REM and the spherical model.
18. Jiří Černý, Véronique Gayrard: Hitting time of large subsets of the hypercube.
Random Structures and Algorithms 33, No. 2, 252-267 (2008).
[ Abstract, pdf, printed version ]
We study the simple random walk on the n-dimensional hypercube, in particular its hitting times of large (possibly random) sets. We give simple conditions on these sets ensuring that the properly-rescaled hitting time is asymptotically exponentially distributed, uniformly in the starting position of the walk. These conditions are then verified for percolation clouds with densities that are much smaller than (n log n)−1. A main motivation behind this paper is the study of the so-called aging phenomenon in the Random Energy Model (REM), the simplest model of a mean-field spin glass. Our results allow us to prove aging in the REM for all temperatures, thereby extending earlier results to their optimal temperature domain.
19. Anton Bovier, Jiří Černý: Hydrodynamic limit for the A+B→∅ model.
Markov Processes and Related Fields 13, No. 3, 543-564 (2007).
[ Abstract, pdf ]
We study a two-species interacting particle model on a subset of Z with open boundaries. The two species are injected with time dependent rate on the left, resp. right boundary. Particles of different species annihilate when they try to occupy the same site. This model has been proposed as a simple model for the dynamics of an "order book" on a stock market. We consider the hydrodynamic scaling limit for the empirical process and prove a large deviation principle that implies convergence to the solution of a non-linear parabolic equation.
20. Gérard Ben Arous, Jiří Černý: Scaling limit for trap models on Zd.
Annals of Probability 35, No. 6, 2356-2385 (2007).
[ Abstract, pdf, printed version ]
We give the "quenched" scaling limit of Bouchaud's trap model in d ≥ 2. This scaling limit is the Fractional-Kinetics process, that is the time change of a d-dimensional Brownian motion by the inverse of an independent α-stable subordinator.
21. Jiří Černý: Moments and distribution of the local time of a two-dimensional random walk.
Stochastic Processes and their Applications 117, No. 2, 262-270 (2007).
[ Abstract, pdf, printed version ]
Let (n,x) be the local time of a random walk on Z2. We prove a strong law of large numbers for the quantity
Ln(α) = ∑xZ2 (n,x)α
for all α ≧ 0. We use this result to describe the distribution of the local time of a typical point in the range of the random walk.
22. Gérard Ben Arous, Jiří Černý: The arcsine law as a universal aging scheme for trap models.
Communications on Pure and Applied Mathematics 61, No. 3, 289-329 (2008).
[ Abstract, pdf, printed version ]
We give a general proof of aging for trap models using the arcsine law for stable subordinators. This proof is based on abstract conditions on the potential theory of the underlying graph and on the randomness of the trapping landscape. We apply this proof to aging for trap models on large two-dimensional tori and for trap dynamics of the Random Energy Model on a broad range of time scales.
23. Gérard Ben Arous, Jiří Černý: Dynamics of trap models.
in Mathematical statistical physics, volume LXXXIII of Les Houches summer school proceedings (2006).
[ Abstract, pdf, printed version ]
These notes cover one of the topics of the class given in the Les Houches Summer School "Mathematical statistical physics" in July 2005. The lectures tried to give a summary of the recent mathematical results about the long-time behaviour of dynamics of (mean-field) spin-glasses and other disordered media. We have chosen here to restrict the scope of these notes to the dynamics of trap models only, but to cover this topic in somewhat more depth.
24. Jiří Černý: The behaviour of aging functions in one-dimensional Bouchaud's trap model.
Communications in Mathematical Physics 261, No. 1, 195-224 (2006).
[ Abstract, pdf, printed version ]
Let τx be a collection of i.i.d. positive random variables with distribution in the domain of attraction of α-stable law with α < 1. The symmetric Bouchaud's trap model on Z is a Markov chain X(t) whose transition rates are given by wxy = (2τx)-1 if x, y are neighbours in Z. We study the behaviour of two correlation functions: P[X(tw+t) = X(tw)] and P[X(t') = X(tw) ∀t'∈[tw,tw+t]]. It is well known that for any of these correlation functions a time-scale t = f(tw) such that aging occurs can be found. We study these correlation functions on time-scales different from f(tw), and we describe more precisely the behaviour of a singular diffusion obtained as the scaling limit of Bouchaud's trap model.
25. Anton Bovier, Jiří Černý, Ostap Hryniv: The Opinion Game: Stock price evolution from microscopic market modelling.
International Journal of Theoretical and Applied Finance (IJTAF) 9, No. 1, 91-112 (2006).
[ Abstract, pdf, printed version ]
We propose a class of Markovian agent based models for the time evolution of a share price in an interactive market. The models rely on a microscopic description of a market of buyers and sellers who change their opinion about the stock value in a stochastic way. The actual price is determined in realistic way by matching (clearing) offers until no further transactions can be performed. Some analytic results for a non-interacting model are presented. We also propose basic interaction mechanisms and show in simulations that these already reproduce certain particular features of prices in real stock markets.
26. Gérard Ben Arous, Jiří Černý, Thomas Mountford: Aging in two-dimensional Bouchaud's model.
Probability Theory and Related Fields 134, No. 1, 1-43 (2006).
[ Abstract, pdf, printed version ]
Let Ex be a collection of i.i.d. exponential random variables. Symmetric Bouchaud's model on Z2 is a Markov chain X(t) whose transition rates are given by wxy= ν exp(βEx) if x, y are neighbours in Z2. We study the behaviour of two correlation functions: P[X(tw+t) = X(tw)] and P[X(t') = X(tw) ∀t'∈[tw,tw+t]]. We prove the (sub)aging behaviour of these functions when β > 1.
27. Gérard Ben Arous, Jiří Černý: Bouchaud's model exhibits two different aging regimes in dimension one.
Annals of Applied Probability 15, No. 2, 1161-1192 (2005).
[ Abstract, pdf ]
Let Ei be a collection of i.i.d. exponential random variables. Bouchaud's model on Z is a Markov chain X(t) whose transition rates are given by wij= exp(-β((1-a)Ei-aEj)) if i, j are neighbours in Z. We study the behaviour of two correlation functions:
P[X(tw+t) = X(tw)]
and
P[X(t') = X(tw) ∀t'∈[tw, tw+t]].
We prove the (sub)aging behaviour of these functions when β > 1 and a∈[0,1].
28. Jiří Černý: Critical Path Analysis for Continuum Percolation.
Annales de l'Institut Henri Poincare (B) Probability and Statistics 40, No. 6, 661-675 (2004).
[ Abstract, pdf, printed version ]
We prove the validity of the Critical path analysis for a continuum percolation model close to Golden-Kozlov one. This is obtained in the limit of strong disorder.
29. Jiří Černý, Roman Kotecký: Interfaces for random cluster models.
J. Statist. Physics 111, No. 1-2, 73-106 (2003).
[ Abstract, printed version ]
A random cluster measure on Zd that is not translationally invariant is constructed for d > 2, the critical density pc, and sufficiently large q. The resulting measure is proven to be a Gibbs state satisfying cluster model DLR-equations.

### Edited monographs

• Random walks, random fields, and disordered systems.
Lecture Notes in Mathematics 2144, Springer, 2015.
edited by Marek Biskup, Jiří Černý and Roman Kotecký. [ printed version ]

### Theses

• Jiří Černý: The existence of translation non-invariant measure in random-cluster model.
Master's thesis. Universita Karlova, Praha, 1998
[ pdf ]
• Jiří Černý: On two properties of strongly disordered systems, Aging and Critical Path Analysis.
PhD thesis, EPF Lausanne, 2003.
[ pdf ]