Zero-temperature Glauber dynamics on the 3-regular tree and the median process
- PDF / 574,543 Bytes
- 44 Pages / 439.37 x 666.142 pts Page_size
- 22 Downloads / 167 Views
Zero-temperature Glauber dynamics on the 3-regular tree and the median process Michael Damron1 · Arnab Sen2 Received: 1 September 2019 / Revised: 28 February 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract In zero-temperature Glauber dynamics, vertices of a graph are given i.i.d. initial spins σx (0) from {−1, +1} with P p (σx (0) = +1) = p, and they update their spins at the arrival times of i.i.d. Poisson processes to agree with a majority of their neighbors. We study this process on the 3-regular tree T3 , where it is known that the critical threshold pc , below which P p -a.s. all spins fixate to −1, is strictly less than 1/2. Defining θ ( p) to be the P p -probability that a vertex fixates to +1, we show that θ is a continuous function on [0, 1], so that, in particular, θ ( pc ) = 0. To do this, we introduce a new continuous-spin process we call the median process, which gives a coupling of all the measures P p . Along the way, we study the time-infinity agreement clusters of the median process, show that they are a.s. finite, and deduce that all continuous spins flip finitely often. In the second half of the paper, we show a correlation decay statement for the discrete spins under P p for a.e. value of p. The proof relies on finiteness of a vertex’s “trace” in the median process to derive a stability of discrete spins under finite resampling. Last, we use our methods to answer a question of Howard (J Appl Probab 37:736–747, 2000) on the emergence of spin chains in T3 in finite time. Keywords Majority vote model · Median process · Zero-temperature Glauber dynamics · Invariant percolation · Mass transport principle Mathematics Subject Classification 60K35 · 82C22
The research of MD is supported by an NSF CAREER Grant. The research of AS is partially supported by NSF DMS 1406247.
B
Michael Damron [email protected]
1
Georgia Tech, Atlanta, USA
2
University of Minnesota, Minneapolis, USA
123
M. Damron, A. Sen
1 Introduction 1.1 The model We study the majority vote model, known as zero-temperature Ising Glauber dynamics, on T3 , the infinite 3-regular tree with vertex set V and edge set E. This is a continuoustime Markov process whose state space is {−1, +1}V , with vertices updating their values at times according to rate-one Poisson clocks to agree with a majority of their neighbors. We take an initial spin configuration σ (0) = (σx (0))x∈V ∈ {−1, +1}V distributed according to the i.i.d. Bernoulli product measure μ p , p ∈ [0, 1], where μ p (σx (0) = +1) = p = 1 − μ p (σx (0) = −1). The configuration σ (t) evolves as t increases according to the zero-temperature limit of Glauber dynamics. To describe this, define the energy (or local cost function) of a vertex x at time t as ex (t) = −
σx (t)σ y (t),
y∈∂ x
where ∂ x is the set of three neighbors of x in T3 . Note that ex (t) is the number of neighbors of x that disagree with x minus the number of neighbors that agree with x at time t. Each vertex has an exponential clock of rate 1 and clocks at different vertices are
Data Loading...
