arXiv subjects

Jiří Černý

Publications and source records attributed to Jiří Černý.

At least 19 recordsLinked to original sources

arXiv2026

Linear spreading speed in non-monotone population models

For a broad class of discrete-time, finite-range interacting particle systems on $\mathbb{Z}$, we establish a linear spreading speed and a one-dimensional shape theorem on the event of survival, without assuming monotonicity or attractiveness of the dynamics. The method requires that the system admits a coupling with supercritical oriented percolation on a coarse-grained lattice. The central technical step is an approximate subadditivity property for the hitting times, obtained through a `shifted coupling' that compensates for the absence of monotonicity. As a concrete application, we show that a discrete-time branching annihilating random walk fits into this framework, and consequently exhibits a linear spreading speed.

math.PR↗

arXiv2026

Tightness of the maximum of branching random walk in random environment and zero-crossings of solutions to discrete parabolic differential equations

We study branching random walk on $\mathbb{Z}$ in a bounded i.i.d. random environment. For this process, we prove that, for almost every realization of the environment, the distributions of the maximally displaced particle (re-centered around their medians) are tight. This extends the result of arXiv:2408.01555 , where tightness was established in the annealed sense, and of arXiv:2212.12390 , where a similar quenched result was proved for branching Brownian motion in random environment. Our proof relies on studying certain discrete-space linear PDEs and establishing that the number of zero-crossings of their solutions is non-increasing in time. We observe that our technique requires no additional assumptions on the environment, in contrast to arXiv:2408.01555 , arXiv:2212.12390 .

math.PR↗

arXiv2025

Scaling limits for the critical level-set percolation of the Gaussian free field on regular trees

We continue the study of the level-set percolation of the discrete Gaussian free field (GFF) on regular trees in the critical regime, initiated in arXiv:2302.02753. First, we derive a sharp asymptotic estimate for the probability that the connected component of the critical level set containing the root of the tree reaches generation $n$. In particular, we show that the one-arm exponent satisfies $ρ=1$. Next, we establish a Yaglom-type limit theorem for the values of the GFF at generation $n$ within this component. Finally, we show that, after a correct rescaling, this component conditioned on reaching generation $n$ converges, as $n\to\infty$, to Aldous' continuum random tree.

math.PR↗

arXiv2025

Canonical Quantization of Spherically Symmetric Einstein-Scalar Field Solutions

We investigate canonical quantization of a general spherically symmetric spacetimes with a massless scalar-field source and examine the associated constraint algebra. The spacetimes are quantized using Dirac's quantization method for constrained systems, producing a set of Wheeler-DeWitt equations. A general solution for some of these equations, specifically the momentum constraint, is found and discussed. A complete solution to the whole set of equations (namely the Hamiltonian constraint) remains yet to be found, however. A full solution can be recovered for a static version of these spacetimes.

gr-qc↗

arXiv2024

On the tightness of the maximum of branching Brownian motion in random environment

We consider one-dimensional branching Brownian motion in spatially random branching environment (BBMRE) and show that for almost every realisation of the environment, the distributions of the maximal particle of the BBMRE re-centred around its median are tight as time evolves. This result is in stark contrast to the fact that the transition fronts in the solution to the randomised Fisher--Kolmogorov--Petrovskii--Piskunov (F-KPP) equation are, in general, not bounded uniformly in time. In particular, this highlights that -- when compared to the settings of homogeneous branching Brownian motion and the F-KPP equation in a homogeneous environment -- the introduction of a random environment leads to a much more intricate behaviour.

math.PR↗

arXiv2024

Survival and complete convergence for a branching annihilating random walk

We study a discrete-time branching annihilating random walk (BARW) on the $d$-dimensional lattice. Each particle produces a Poissonian number of offspring with mean $μ$ which independently move to a uniformly chosen site within a fixed distance $R$ from their parent's position. Whenever a site is occupied by at least two particles, all the particles at that site are annihilated. We prove that for any $μ>1$ the process survives when $R$ is sufficiently large. For fixed $R$ we show that the process dies out if $μ$ is too small or too large. Furthermore, we exhibit an interval of $μ$-values for which the process survives and possesses a unique non-trivial ergodic equilibrium for $R$ sufficiently large. We also prove complete convergence for that case.

math.PR↗

arXiv2023

Critical and near-critical level-set percolation of the Gaussian free field on regular trees

For the Gaussian free field on a $(d + 1)$-regular tree with $d \geq 2$, we study the percolative properties of its level sets in the critical and the near-critical regime. In particular, we show the continuity of the percolation probability, derive an exact symptotic tail estimate for the cardinality of the connected component of the critical level set, and describe the asymptotic behaviour of the percolation probability in the near-critical regime.

math.PR↗

arXiv2021

Giant component for the supercritical level-set percolation of the Gaussian free field on regular expander graphs

We consider the zero-average Gaussian free field on a certain class of finite $d$-regular graphs for fixed $d\ge 3$. This class includes $d$-regular expanders of large girth and typical realisations of random $d$-regular graphs. We show that the level set of the zero-average Gaussian free field above level $h$ has a giant component in the whole supercritical phase, that is for all $h<h_\star$, with probability tending to one as the size of the graphs tends to infinity. In addition, we show that this component is unique. This significantly improves the result of [AC20b], where it was shown that a linear fraction of vertices is in mesoscopic components if $h<h_\star$.

math.PR↗

arXiv2021

Triviality of the geometry of mixed $p$-spin spherical Hamiltonians with external field

We study isotropic Gaussian random fields on the high-dimensional sphere with an added deterministic linear term, also known as mixed p-spin Hamiltonians with external field. We prove that if the external field is sufficiently strong, then the resulting function has trivial geometry, that is only two critical points. This contrasts with the situation of no or weak external field where these functions typically have an exponential number of critical points. We give an explicit threshold $h_c$ for the magnitude of the external fieldnecessary for trivialization and conjecture $h_c$ to be sharp. The Kac-Rice formula is our main tool. Our work extends [Fyo15], which identified the trivial regime for the special case of pure p-spin Hamiltonians with random external field.

math.PR↗

arXiv2021

(Un-)bounded transition fronts for the parabolic Anderson model and the randomized F-KPP equation

We investigate the uniform boundedness of the fronts of the solutions to the randomized Fisher-KPP equation and to its linearization, the parabolic Anderson model. It has been known that for the standard (i.e. deterministic) Fisher-KPP equation, as well as for the special case of a randomized Fisher-KPP equation with so-called ignition type nonlinearity, one has a uniformly bounded (in time) transition front. Here, we show that this property of having a uniformly bounded transition front fails to hold for the general randomized Fisher-KPP equation. Nevertheless, we establish that this property does hold true for the parabolic Anderson model.

math.AP↗

arXiv2019

Level-set percolation of the Gaussian free field on regular graphs II: Finite expanders

We consider the zero-average Gaussian free field on a certain class of finite $d$-regular graphs for fixed $d\geq 3$. This class includes $d$-regular expanders of large girth and typical realisations of random $d$-regular graphs. We show that the level set of the zero-average Gaussian free field above level $h$ exhibits a phase transition at level $h_\star$, which agrees with the critical value for level-set percolation of the Gaussian free field on the infinite $d$-regular tree. More precisely, we show that, with probability tending to one as the size of the finite graphs tends to infinity, the level set above level $h$ does not contain any connected component of larger than logarithmic size whenever $h>h_\star$, and on the contrary, whenever $h<h_\star$, a linear fraction of the vertices is contained in connected components of the level set above level $h$ having a size of at least a small fractional power of the total size of the graph. It remains open whether in the supercritical phase $h<h_\star$, as the size of the graphs tends to infinity, one observes the emergence of a (potentially unique) giant connected component of the level set above level $h$. The proofs in this article make use of results from the accompanying paper [AC1].

math.PR↗

arXiv2019

Level-set percolation of the Gaussian free field on regular graphs I: Regular trees

We study level-set percolation of the Gaussian free field on the infinite $d$-regular tree for fixed $d\geq 3$. Denoting by $h_\star$ the critical value, we obtain the following results: for $h>h_\star$ we derive estimates on conditional exponential moments of the size of a fixed connected component of the level set above level $h$; for $h<h_\star$ we prove that the number of vertices connected over distance $k$ above level $h$ to a fixed vertex grows exponentially in $k$ with positive probability. Furthermore, we show that the percolation probability is a continuous function of the level $h$, at least away from the critical value $h_\star$. Along the way we also obtain matching upper and lower bounds on the eigenfunctions involved in the spectral characterisation of the critical value $h_\star$ and link the probability of a non-vanishing limit of the martingale used therein to the percolation probability. A number of the results derived here are applied in the accompanying paper [AC2].

math.PR↗

arXiv2019

Quenched invariance principles for the maximal particle in branching random walk in random environment and the parabolic Anderson model

We consider branching random walk in spatial random branching environment (BRWRE) in dimension one, as well as related differential equations: the Fisher-KPP equation with random branching and its linearized version, the parabolic Anderson model (PAM). When the random environment is bounded, we show that after recentering and scaling, the position of the maximal particle of the BRWRE, the front of the solution of the PAM, as well as the front of the solution of the randomized Fisher-KPP equation fulfill quenched invariance principles. In addition, we prove that at time t the distance between the median of the maximal particle of the BRWRE and the front of the solution of the PAM is in O(ln t). This partially transfers results from Bramson [Comm. Pure Appl. Math. 31 (1978), no. 5, 531--581] to the setting of BRWRE.

math.PR↗

arXiv2018

Markovian dynamics of exchangeable arrays

We study Markov processes with values in the space of general two-dimensional arrays whose distribution is exchangeable. The results of this paper are inspired by the theory of exchangeable dynamical random graphs developed by H. Crane (2016, 2017).

math.PR↗

arXiv2016

Random walks in dynamic random environments and ancestry under local population regulation

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 spatial embeddings of ancestral lineages 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.

math.PR↗

arXiv2015

Mixing time for the random walk on the range of the random walk on tori

Consider the subgraph of the discrete $d$-dimensional torus of size length $N$, $d\ge3$, induced by the range of the simple random walk on the torus run until the time $uN^d$. We prove that for all $d\ge 3$ and $u>0$, the mixing time for the random walk on this subgraph is of order $N^2$ with probability at least $1 - Ce^{-(\log N)^2}$.

math.PR↗

arXiv2015

Randomly trapped random walks

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 $\mathbb {Z}$. 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.

math.PR↗

arXiv2015

Aging of the Metropolis dynamics on the Random Energy Model

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.

math.PR↗