arXiv subjects

Shirshendu Chatterjee

Publications and source records attributed to Shirshendu Chatterjee.

24 records · Page 2Linked to original sources

arXiv2015

Multiple phase transitions in long-range first-passage percolation on square lattices

We consider a model of long-range first-passage percolation on the $d$ dimensional square lattice $Z^d$ in which any two distinct vertices $x, y \in Z^d$ are connected by an edge having exponentially distributed passage time with mean $||x-y||^{α+o(1)}$, where $α>0$ is a fixed parameter and $||\cdot||$ is the $\ell_1$-norm on $Z^d$. We analyze the asymptotic growth rate of the set $B_t$, which consists of all $x \in Z^d$ such that the first-passage time between the origin 0 and $x$ is at most $t$, as $t\to\infty$. We show that depending on the values of $α$ there are four growth regimes: (i) instantaneous growth for $α 2d+1$ like the nearest-neighbor first-passage percolation model corresponding to $α=\infty$.

math.PR↗

arXiv2013

A first order phase transition in the threshold-$θ\ge 2$ contact process on random $r$-regular graphs and $r$-trees

We consider the discrete-time threshold-$θ\ge 2$ contact process on a random r-regular graph on n vertices. In this process, a vertex with at least θoccupied neighbors at time t will be occupied at time t+1 with probability p, and vacant otherwise. We show that if $θ\ge 2$ and $r \ge θ+2$, $ε_1$ is small and p is at least $p_1(ε_1)$, then starting from all vertices occupied the fraction of occupied vertices stays above $1-2ε_1$ up to time $\exp(γ_1(r)n)$ with probability at least $1 - \exp(-γ_1(r)n)$. In the other direction, we show that for $p_2 < 1$ there is an $ε_2(p_2)>0$ so that if $p \le p_2$ and the number of occupied vertices in the initial configuration is at most $ε_2(p_2)n$, then with high probability all vertices are vacant at time $C_2(p_2) \log(n)$. These two conclusions imply that on the random r-regular graph there cannot be a quasi-stationary distribution with density of occupied vertices between 0 and $ε_2(p_1)$, and allow us to conclude that the process on the r-tree has a first order phase transition.

math.PR↗

arXiv2013

The order-chaos phase transition for a general class of complex Boolean networks

We consider a model for heterogeneous 'gene regulatory networks' that is a generalization of the model proposed by Chatterjee and Durrett (2011) as an "annealed approximation" of Kauffmann's (1969) random Boolean networks. In this model, genes are represented by the nodes of a random directed graph on n vertices with specified in-degree distribution (resp. out-degree distribution or joint distribution of in-degree and out-degree), and the expression bias (the expected fraction of 1's in the Boolean functions) p is same for all nodes. Following a standard practice in the physics literature, we use a discrete-time threshold contact process with parameter q=2p(1-p) (in which a vertex with at least one 'occupied' input at time t will be occupied at time t+1 with probability q, and 'vacant' otherwise) on the above random graph to approximate the dynamics of the Boolean network. We show that there is a parameter r (which can be written explicitly in terms of first few moments of the degree distribution) such that, with probability tending to 1 as n goes to infinity, if 2p(1-p)r>1, then starting from all occupied sites the threshold contact process maintains a positive ({\it quasi-stationary}) density of occupied sites for time which is exponential in n, whereas if 2p(1-p)r<1, then the persistence time of the threshold contact process is at most logarithmic in n. These two phases correspond to the 'chaotic' and 'ordered' behavior of the gene networks.

math.PR↗

arXiv2012

Asymptotic behavior of Aldous' gossip process

Aldous [(2007) Preprint] defined a gossip process in which space is a discrete $N\times N$ torus, and the state of the process at time $t$ is the set of individuals who know the information. Information spreads from a site to its nearest neighbors at rate 1/4 each and at rate $N^{-α}$ to a site chosen at random from the torus. We will be interested in the case in which $α<3$, where the long range transmission significantly accelerates the time at which everyone knows the information. We prove three results that precisely describe the spread of information in a slightly simplified model on the real torus. The time until everyone knows the information is asymptotically $T=(2-2α/3)N^{α/3}\log N$. If $ρ_s$ is the fraction of the population who know the information at time $s$ and $\varepsilon$ is small then, for large $N$, the time until $ρ_s$ reaches $\varepsilon$ is $T(\varepsilon)\approx T+N^{α/3}\log (3\varepsilon /M)$, where $M$ is a random variable determined by the early spread of the information. The value of $ρ_s$ at time $s=T(1/3)+tN^{α/3}$ is almost a deterministic function $h(t)$ which satisfies an odd looking integro-differential equation. The last result confirms a heuristic calculation of Aldous.

math.PR↗

arXiv2009

Contact processes on random graphs with power law degree distributions have critical value 0

If we consider the contact process with infection rate $λ$ on a random graph on $n$ vertices with power law degree distributions, mean field calculations suggest that the critical value $λ_c$ of the infection rate is positive if the power $α>3$. Physicists seem to regard this as an established fact, since the result has recently been generalized to bipartite graphs by Gómez-Gardeñes et al. [Proc. Natl. Acad. Sci. USA 105 (2008) 1399--1404]. Here, we show that the critical value $λ_c$ is zero for any value of $α>3$, and the contact process starting from all vertices infected, with a probability tending to 1 as $n\to\infty$, maintains a positive density of infected sites for time at least $\exp(n^{1-δ})$ for any $δ>0$. Using the last result, together with the contact process duality, we can establish the existence of a quasi-stationary distribution in which a randomly chosen vertex is occupied with probability $ρ(λ)$. It is expected that $ρ(λ)\sim Cλ^β$ as $λ\to0$. Here we show that $α-1\leβ\le2α-3$, and so $β>2$ for $α>3$. Thus even though the graph is locally tree-like, $β$ does not take the mean field critical value $β=1$.

math.PR↗

arXiv2009

Persistence of Activity in Threshold Contact Processes, an "Annealed Approximation" of Random Boolean Networks

We consider a model for gene regulatory networks that is a modification of Kauffmann's (1969) random Boolean networks. There are three parameters: $n =$ the number of nodes, $r =$ the number of inputs to each node, and $p =$ the expected fraction of 1's in the Boolean functions at each node. Following a standard practice in the physics literature, we use a threshold contact process on a random graph on $n$ nodes, in which each node has in degree $r$, to approximate its dynamics. We show that if $r\ge 3$ and $r \cdot 2p(1-p)>1$, then the threshold contact process persists for a long time, which correspond to chaotic behavior of the Boolean network. Unfortunately, we are only able to prove the persistence time is $\ge \exp(cn^{b(p)})$ with $b(p)>0$ when $r\cdot 2p(1-p)> 1$, and $b(p)=1$ when $(r-1)\cdot 2p(1-p)>1$.

math.PR↗