85 citations · 323 across the 14 of their papers we have counts for
17 papers
Freiheitssatz and phase transition for the density model of random groups
Tsung-Hsuan Tsai
Magnus' Freiheitssatz states that if a group is defined by a presentation with generators and a single relator containing the last generating letter, then the first lette…
Andrásfai and Vega graphs in Ramsey-Turán theory
Tomasz Łuczak, Joanna Polcyn, Christian Reiher
Given positive integers , we let denote the maximum number of edges in a triangle-free graph on vertices with . In the early sixties…
On the Ramsey-Turán density of triangles
Tomasz Łuczak, Joanna Polcyn, Christian Reiher
One of the oldest results in modern graph theory, due to Mantel, asserts that every triangle-free graphs on vertices has at most edges. About half a cent…
Integral homology of random simplicial complexes
Tomasz Łuczak, Yuval Peled
The random -dimensional simplicial complex process starts with a complete graph on vertices, and in every step a new -dimensional face, chosen uniformly at random, is add…
Asymmetry and structural information in preferential attachment graphs
Tomasz Luczak, Abram Magner, Wojciech Szpankowski
Graph symmetries intervene in diverse applications, from enumeration, to graph structure compression, to the discovery of graph dynamics (e.g., node arrival order inference). Where…
Manipulative waiters with probabilistic intuition
Mał gorzata Bednarska-Bzdȩga, Dan Hefetz, Michael Krivelevich +1
For positive integers and and a monotone graph property $\cA$, we consider the two player, perfect information game $\WC(n,q,\cA)$, which is defined as follows. The game pr…
A sharp threshold for collapse of the random triangular group
Sylwia Antoniuk, Ehud Friedgut, Tomasz Łuczak
The random triangular group is the group given by a random group presentation with generators in which every relator of length three is present independently with prob…
Random triangular groups at density 1/3
Sylwia Antoniuk, Tomasz Łuczak, Jacek Świcatkowski
Let Γ(n,p) denote the binomial model of a random triangular group. We show that there exist constants c, C > 0 such that if p <= c/n^2, then a.a.s. Γ(n,p) is free and if p >= C log…
Collapse of random triangular groups: a closer look
Sylwia Antoniuk, Tomasz Łuczak, Jacek Świcatkowski
The random triangular group Γ(n,t) is a group given by a presentation P=<S|R>, where S is a set of n generators and R is a random set of t cyclically reduced words of length three.…
Trapping a particle of a quantum walk on the line
Antoni Wojcik, Tomasz Luczak, Pawel Kurzynski +3
We observe that changing a phase at a single point in a discrete quantum walk results in a rather surprising localization effect. For certain values of this phase change the possib…
Limit measures of inhomogeneous discrete-time quantum walks in one dimension
Norio Konno, Tomasz Luczak, Etsuo Segawa
We treat three types of measures of the quantum walk (QW) with the spatial perturbation at the origin, which was introduced by [1]: time averaged limit measure, weak limit measure,…
Coloring dense graphs via VC-dimension
Tomasz Łuczak, Stéphan Thomassé
The Vapnik-Červonenkis dimension is a complexity measure of set-systems, or hypergraphs. Its application to graphs is usually done by considering the sets of neighborhoods of the v…
Two critical periods in the evolution of random planar graphs
Mihyun Kang, Tomasz Łuczak
Let be a graph chosen uniformly at random from the family of all labeled planar graphs with vertices and edges. In the paper we study the component structure of $P…
On the Multi-coloured Ramsey Numbers of Cycles
Tomasz Łuczak, Miklós Simonovits, Jozef Skokan
For a graph and an integer , denotes the smallest integer for which for any edge-colouring of the complete graph by colours there exists a colou…
The Ramsey number for a triple of large cycles
Agnieszka Figaj, Tomasz Luczak
We find the asymptotic value of the Ramsey number for a triple of long cycles, where the lengths of the cycles are large but may have different parity.
Multiuser quantum communication networks
A. Wojcik, T. Luczak, P. Kurzynski +3
We study a quantum state transfer between spins interacting with an arbitrary network of spins coupled by uniform XX interactions. It is shown that in such a system under fairly ge…
Quantum walks on cycles
Malgorzata Bednarska, Andrzej Grudka, Pawel Kurzynski +2
We consider asymptotic behaviour of a Hadamard walk on a cycle. For a walk which starts with a state in which all the probability is concentrated on one node, we find the explicit…