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View article: Chemical distance in graphs of polynomial growth
Chemical distance in graphs of polynomial growth Open
We prove an Antal-Pisztora type theorem for transitive graphs of polynomial growth. That is, we show that if $G$ is a transitive graph of polynomial growth and $p > p_c(G)$, then for any two sites $x, y$ of $G$ which are connected by a $p$…
View article: On one-dimensional Cluster cluster model
On one-dimensional Cluster cluster model Open
The Cluster-cluster model was introduced by Meakin et al in 1984. Each $x\in \mathbb{Z}^d$ starts with a cluster of size 1 with probability $p \in (0,1]$ independently. Each cluster $C$ performs a continuous-time SRW with rate $|C|^{-α}$. …
View article: Minimal harmonic measure on 2D lattices
Minimal harmonic measure on 2D lattices Open
We study the harmonic measure (i.e. the limit of the hitting distribution of a simple random walk starting from a distant point) on three canonical two-dimensional lattices: the square lattice $$\mathbb {Z}^2$$ , the triangular lattic…
View article: Logarithmic fluctuations of Stationary Hastings-Levitov
Logarithmic fluctuations of Stationary Hastings-Levitov Open
We prove that the fluctuation field $\{M_t(x)\}_{x\in\mathbb{R}}$ of stationary Hastings-Levitov$(0)$ exhibits logarithmic spatial correlations. Moreover, by studying the infinitesimal generator of the imaginary part of $M_t(0)$, we show t…
View article: Minimal harmonic measure on 2D lattices
Minimal harmonic measure on 2D lattices Open
We study the harmonic measure (i.e. the limit of the hitting distribution of a simple random walk starting from a distant point) on three canonical two-dimensional lattices: the square lattice $\mathbb{Z}^2$, the triangular lattice $\maths…
View article: The Double Bubble Problem in the Hexagonal Norm
The Double Bubble Problem in the Hexagonal Norm Open
We study the double bubble problem where the perimeter is taken with respect to the hexagonal norm, i.e. the norm whose unit circle in $\mathbb{R}^2$ is the regular hexagon. We provide an elementary proof for the existence of minimizing se…
View article: Vertex-removal stability and the least positive value of harmonic measures
Vertex-removal stability and the least positive value of harmonic measures Open
We prove that for $\mathbb{Z}^d$ ($d\ge 2$), the vertex-removal stability of harmonic measures (i.e. it is feasible to remove some vertex while changing the harmonic measure by a bounded factor) holds if and only if $d=2$. The proof mainly…
View article: Cylindrical Hastings Levitov
Cylindrical Hastings Levitov Open
We define a Hastings-Levitov$(0)$ process on a cylinder and prove that the process converges to Stationary Hastings Levitov$(0)$ under appropriate particle size scaling that depends on the radius of the cylinder. The Stationary Hastings Le…
View article: Growth of stationary Hastings–Levitov
Growth of stationary Hastings–Levitov Open
We construct and study a stationary version of the Hastings-Levitov$(0)$\nmodel. We prove that, unlike in the classical HL$(0)$ model, in the stationary\ncase the size of particles attaching to the aggregate is tight, and therefore\nSHL$(0…
View article: A toy model for DLA arm growth in a wedge
A toy model for DLA arm growth in a wedge Open
In this paper, we consider a non-homogeneous discrete-time Markov chain which can be seen as a toy model for the growth of the arms of the DLA (Diffusion limited aggregation) process in a sub-linear wedge. It is conjectured that in a thin …
View article: Continuity and uniqueness of percolation critical parameters in finitary random interlacements
Continuity and uniqueness of percolation critical parameters in finitary random interlacements Open
We prove that the critical percolation parameter for Finitary Random Interlacements (FRI) is continuous with respect to the path length parameter T. The proof uses a result which is interesting on its own right; equality of natural critica…
View article: The chemical distance in random interlacements in the low-intensity regime
The chemical distance in random interlacements in the low-intensity regime Open
In $\mathbb{Z}^d$ with $d\ge 5$, we consider the time constant $ρ_u$ associated to the chemical distance in random interlacements at low intensity $u \ll 1$. We prove an upper bound of order $u^{-1/2}$ and a lower bound of order $u^{-1/2+\…
View article: Discrete $\ell^{1}$ Double Bubble solution is at most ceiling +2 of the continuous solution
Discrete $\ell^{1}$ Double Bubble solution is at most ceiling +2 of the continuous solution Open
In this paper we show that the solution of the discrete Double Bubble problem over $\mathbb{Z}^2$ is at most the ceiling function plus two of the continuous solution to the Double Bubble problem, with respect to the $\ell^1$ norm, found in…
View article: Continuity and uniqueness of percolation critical parameters in Finitary Random Interlacements
Continuity and uniqueness of percolation critical parameters in Finitary Random Interlacements Open
We prove that the critical percolation parameter for Finitary Random Interlacements (FRI) is continuous with respect to the path length parameter $T$. The proof uses a result which is interesting on its own right; equality of natural criti…
View article: Dimension of diffusion-limited aggregates grown on a line
Dimension of diffusion-limited aggregates grown on a line Open
Diffusion-limited aggregation (DLA) has served for 40 years as a paradigmatic example for the creation of fractal growth patterns. In spite of thousands of references, no exact result for the fractal dimension D of DLA is known. In this Le…
View article: An elementary proof for the Double Bubble problem in $\ell^1$ norm
An elementary proof for the Double Bubble problem in $\ell^1$ norm Open
We study the double bubble problem with perimeter taken with respect to the $\ell_1$ norm on $\mathbb{R}^2$. We give an elementary proof for the existence of minimizing sets for any volume ratio parameter $0
View article: Growth of Stationary Hastings-Levitov
Growth of Stationary Hastings-Levitov Open
We construct and study a stationary version of the Hastings-Levitov$(0)$ model. We prove that, unlike in the classical HL$(0)$ model, in the stationary case the size of particles attaching to the aggregate is tight, and therefore SHL$(0)$ …
View article: On covering monotonic paths with simple random walk
On covering monotonic paths with simple random walk Open
In this paper we study the probability that a $d$ dimensional simple random walk (or the first $L$ steps of it) covers each point in a nearest neighbor path connecting 0 and the boundary of an $L_{1}$ ball. We show that among all such path…
View article: Stabilization of DLA in a wedge
Stabilization of DLA in a wedge Open
We consider Diffusion Limited Aggregation (DLA) in a two-dimensional wedge. We prove that if the angle of the wedge is smaller than $\\pi /4$, there is some $a>2$ such that almost surely, for all $R$ large enough, after time $R^{a}$ all ne…
View article: Scaling limit of DLA on a long line segment
Scaling limit of DLA on a long line segment Open
In this paper, we prove that the bulk of DLA starting from a long line segment on the $x$-axis has a scaling limit to the stationary DLA process (SDLA). The main phenomenological difficulty is the multi-scale, non-monotone interaction of t…
View article: Percolation for the Finitary Random interlacements
Percolation for the Finitary Random interlacements Open
In this paper, we prove a phase transition in the connectivity of Finitary Random interlacements $\mathcal{FI}^{u,T}$ in $\mathbb{Z}^d$, with respect to the average stopping time. For each $u>0$, with probability one $\mathcal{FI}^{u,T}$ h…
View article: Stationary Harmonic Measure as the Scaling Limit of Truncated Harmonic Measure
Stationary Harmonic Measure as the Scaling Limit of Truncated Harmonic Measure Open
In this paper we prove that the stationary harmonic measure of an infinite set in the upper planar lattice can be represented as the proper scaling limit of the classical harmonic measure of truncations of the infinite set.
View article: Stabilization of DLA in a wedge
Stabilization of DLA in a wedge Open
We consider Diffusion Limited Aggregation (DLA) in a two-dimensional wedge. We prove that if the angle of the wedge is smaller than $π/4$, there is some $a>2$ such that almost surely, for all $R$ large enough, after time $R^a$ all new part…
View article: Eigenvalue versus perimeter in a shape theorem for self-interacting random walks
Eigenvalue versus perimeter in a shape theorem for self-interacting random walks Open
We study paths of time-length $t$ of a continuous-time random walk on\n$\\mathbb Z^2$ subject to self-interaction that depends on the geometry of the\nwalk range and a collection of random, uniformly positive and finite edge\nweights. The …
View article: On covering paths with 3 dimensional random walk
On covering paths with 3 dimensional random walk Open
In this paper we find an upper bound for the probability that a $3$ dimensional simple random walk covers each point in a nearest neighbor path connecting 0 and the boundary of an $L_1$ ball of radius $N$ in $\\mathbb{Z} ^d$. For $d\\ge 4$…
View article: Stationary Harmonic Measure and DLA in the Upper half Plane
Stationary Harmonic Measure and DLA in the Upper half Plane Open
In this paper, we introduce the stationary harmonic measure in the upper half plane. By bounding this measure, we are able to define both the discrete and continuous time diffusion limit aggregation (DLA) in the upper half plane with absor…
View article: On sets of zero stationary harmonic measure
On sets of zero stationary harmonic measure Open
In this paper, we prove that any subset with an appropriate sub-linear horizontal growth has a non-zero stationary harmonic measure. On the other hand, we also show any subset with super-linear horizontal growth will have a $0$ stationary …