arXiv subjects

Alexander Shamov

Publications and source records attributed to Alexander Shamov.

7 recordsLinked to original sources

arXiv2026

Non-colliding billiards in the plane

We present an open problem about non-colliding freely moving hard disks in the Euclidean plane, together with related positive and negative partial results. The open problem is stated in a non-degenerate form: velocities are required to be pairwise distinct and their speeds are required to be uniformly bounded away from infinity. The positive deterministic result gives a bounded, injective, non-colliding velocity assignment for the integer lattice; after a common velocity shift, the speeds are also bounded away from zero. The negative result shows that no bounded continuous vector field on the whole plane can serve as a universal assignment satisfying the same separation inequality for all pairs of points at distance greater than one. We also record a space-time interpretation of the problem, relate it to packings by nonparallel cylinders in three dimensions, and formulate a corresponding topological-dynamical question for cylinder packings.

math.DS↗

arXiv2018

Kernels of conditional determinantal measures and the proof of the Lyons-Peres Conjecture

The main result of this paper, Theorem 1.5, establishes a conjecture of Lyons and Peres: for a determinantal point process governed by a reproducing kernel, the system of kernels sampled at the particles of a random configuration is complete in the range of the kernel. A key step in the proof, Lemma 1.11, states that conditioning on the configuration in a subset preserves the determinantal property, and the main Lemma 1.12 is a new local property for kernels of conditional point processes. In Theorem 1.7 we prove the triviality of the tail sigma-algebra for determinantal point processes governed by self-adjoint kernels.

math.PR↗

arXiv2016

Where does a random process hit a fractal barrier?

Given a Brownian path $β(t)$ on $\mathbb{R}$, starting at $1$, a.s. there is a singular time set $T_β$, such that the first hitting time of $β$ by an independent Brownian motion, starting at $0$, is in $T_β$ with probability one. A couple of problems regarding hitting measure for random processes are presented.

math.PR↗

arXiv2016

On Gaussian multiplicative chaos

We propose a new definition of the Gaussian multiplicative chaos (GMC) and an approach based on the relation of subcritical GMC to randomized shifts of a Gaussian measure. Using this relation we prove general uniqueness and convergence results for subcritical GMC that hold for Gaussian fields with arbitrary covariance kernels.

math.PR↗

arXiv2016

Weak and Strong disorder for the stochastic heat equation and the continuous directed polymer in $d\geq 3$

We consider the smoothed multiplicative noise stochastic heat equation $$d u_{\eps,t}= \frac 12 Δu_{\eps,t} d t+ β\eps^{\frac{d-2}{2}}\, \, u_{\eps, t} \, d B_{\eps,t} , \;\;u_{\eps,0}=1,$$ in dimension $d\geq 3$, where $B_{\eps,t}$ is a spatially smoothed (at scale $\eps$) space-time white noise, and $β>0$ is a parameter. We show the existence of a $\barβ\in (0,\infty)$ so that the solution exhibits weak disorder when $β<\barβ$ and strong disorder when $β> \barβ$. The proof techniques use elements of the theory of the Gaussian multiplicative chaos.

math.PR↗

arXiv2015

Bi-Lipschitz bijections of $\mathbb{Z}$

It is shown that every bi-Lipschitz bijection from $\mathbb{Z}$ to itself is at a bounded $L_{\infty}$ distance from either the identity or the reflection. We then comment on the group-theoretic properties of the action of bi-Lipschitz bijections.

math.MG↗

arXiv2010

On short-time asymptotics of one-dimensional Harris flows

We study the short-time asymptotical behavior of stochastic flows on \mathbb{R} in the \sup-norm. The results are stated in terms of a Gaussian process associated with the covariation of the flow. In case the Gaussian process has a continuous version the two processes can be coupled in such a way that the difference is uniformly $o(\ln\ln t^{-1})$. In case it has no continuous version, an $O(\ln\ln t^{-1})$ estimate is obtained under mild regularity assumptions. The main tools are Gaussian measure concentration and a martingale version of the Slepian comparison principle.

math.PR↗