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A stochastic telegraph equation from the six-vertex model
Probability · Submitted 24 Mar 2018
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We show that such fields arise naturally as asymptotic fluctuations of the height function in a certain limit regime of the stochastic six vertex model in a quadrant.
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The abstract, sentence by sentence
A stochastic telegraph equation is defined by adding a random inhomogeneity to the classical (second order linear hyperbolic) telegraph differential equation. The inhomogeneities we consider are proportional to the two-dimensional white noise, and solutions to our equation are two-dimensional random Gaussian fields. We show that such fields arise naturally as asymptotic fluctuations of the height function in a certain limit regime of the stochastic six vertex model in a quadrant. The corresponding law of large numbers -- the limit shape of the height function -- is described by the (deterministic) homogeneous telegraph equation.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | A stochastic telegraph equation is defined by adding a random inhomogeneity to the classical (second order linear hyperbolic) telegraph differential equation. | background 0.52 | 0.45 | 0.07 | asserted 1.00 |
| s1 | The inhomogeneities we consider are proportional to the two-dimensional white noise, and solutions to our equation are two-dimensional random Gaussian fields. | method 0.46 | 0.69 | 0.07 | asserted 1.00 |
| s2 | We show that such fields arise naturally as asymptotic fluctuations of the height function in a certain limit regime of the stochastic six vertex model in a quadrant. | result 0.89 | 0.93 | 0.73 | asserted 1.00 |
| s3 | The corresponding law of large numbers -- the limit shape of the height function -- is described by the (deterministic) homogeneous telegraph equation. | result 0.97 | 0.83 | 0.19 | asserted 1.00 |
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The paper
Original abstract
A stochastic telegraph equation is defined by adding a random inhomogeneity to the classical (second order linear hyperbolic) telegraph differential equation. The inhomogeneities we consider are proportional to the two-dimensional white noise, and solutions to our equation are two-dimensional random Gaussian fields. We show that such fields arise naturally as asymptotic fluctuations of the height function in a certain limit regime of the stochastic six vertex model in a quadrant. The corresponding law of large numbers -- the limit shape of the height function -- is described by the (deterministic) homogeneous telegraph equation.
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How the literature uses it
13 arXiv papers cite this in 36 sentences
Papers citing it · 25 sentences without a confident label
Builds on it 1 paper
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The computation of the covariance for the vectors in [ref] is based on the following relation generalising [ref] .
Shift-invariance for vertex models and polymers · Four point relation for the colored stochastic six-vertex model · 0.80
Uses its method or tool 1 paper
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There are four cases pertaining to such conditions, and in each case $(x+1,y+1)$ is computed in [ref] , using the `four point relation' derived therein.
Stochastic Telegraph Equation Limit for the Stochastic Six Vertex Model · Preliminary · 0.99
Compares with it 1 paper
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[ref] proves a law of large numbers and some Gaussian fluctuation results under this scaling.
Stochastic PDE Limit of the Six Vertex Model · Stochastic six vertex model · 0.91
Cites it as background 4 papers
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Next, in the Gaussian models (additive stochastic heat equation and colored stochastic telegraph equation in Figure [ref]; see Borodin-Gorin [ref] for a description of the telegraph limit in the colorless situation) we noticed that shift-invariance reduces, still nontrivially, to that for local times of Brownian bridges and numbers of intersections of two persistent random walks.
Shift-invariance for vertex models and polymers · Shift-invariance for colored stochastic vertex models · 0.83
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Over the past several years, this model has received considerable interest [ref] in the probability and mathematical physics communities, due to the fact that it is one of the simplest models that lies in the intersection of two-dimensional statistical mechanics and stochastic integrability; as such, it exhibits a number of interesting physical phenomena.
Limit Shapes and Local Statistics for the Stochastic Six-Vertex Model · Preface · 0.97
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There was also a flurry of recent activity in proving different asymptotic results for it, see, e.g., Borodin-Corwin-Gorin [ref] , Aggarwal [ref] , Aggarwal-Borodin [ref] , Reshetikhin-Sridhar [ref] , Corwin-Ghosal-Shen-Tsai [ref] , Borodin [ref] , Borodin-Olshanski [ref] , Borodin-Gorin [ref] , Shen-Tsai [ref] .
Color-position symmetry in interacting particle systems · The symmetry for the stochastic six vertex model · 0.99
What it cites · 34 sentences without a confident label
Builds on 1 paper
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We can even reach the collections of points on more general monotone paths: [equation] for the domain wall boundary conditions this was done in [ref] , and here the situation is analogous.
Between the stochastic six vertex model and Hall-Littlewood processes · Appendix B: Theorem [ref] through a variational principle and contour integrals · 0.84
Uses the method or tool of 5 papers
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We rely on several ingredients – contour integral expressions of [ref] , a Gaussianity lemma for random variables with moments given by contour integrals of [ref] , and a novel combinatorial argument of Theorem [ref] linking cumulants with their shifted versions.
General beta Jacobi corners process and the Gaussian Free Field · Limit regime and main asymptotic results · 0.99
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We rely on several ingredients – contour integral expressions of [ref] , a Gaussianity lemma for random variables with moments given by contour integrals of [ref] , and a novel combinatorial argument of Theorem [ref] linking cumulants with their shifted versions.
Symmetric elliptic functions, IRF models, and dynamic exclusion processes · Limit regime and main asymptotic results · 0.99
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For the computations we rely on contour integral formulas of [ref] .
Current Fluctuations of the Stationary ASEP and Six-Vertex Model · Appendix B: Theorem [ref] through a variational principle and contour integrals · 1.00
Compares with 4 papers
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The expression $q^y-x+1 - α^-1 q^2k-2 -q^k-1 Ø(x,y)$ in [ref] can be written as [equation] In the case $α=0$, the observable $E_N$ simplifies to [equation] The formula [ref] matches [ref] , with $x_1=x_2=…=t+1$, $y=x$.
Stochastic six-vertex model · Observables · 0.90
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On the other hand, we later realized that one of our heuristic arguments could be also turned into a complete proof (different from the one in [ref] ); it is this proof that we include below.
Stochastic Telegraph Equation Limit for the Stochastic Six Vertex Model · CLT for general boundary conditions · 0.87
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One might be surprised that while the six-vertex hydrodynamic equation of [ref] , [ref] does not look linear, [ref] is.
Limit Shapes of the Stochastic Six Vertex Model · Preface · 0.97
Cites as background 20 papers
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More concretely, we deal with the stochastic six-vertex model (as well as its deformation – the dynamic six–vertex model), that was first introduced in [ref] and whose asymptotic behavior has been recently studied in [ref] , [ref] , [ref] , [ref] , [ref] , [ref] , [ref] .
Stochastic six-vertex model · Preface · 0.92
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More concretely, we deal with the stochastic six-vertex model (as well as its deformation – the dynamic six–vertex model), that was first introduced in [ref] and whose asymptotic behavior has been recently studied in [ref] , [ref] , [ref] , [ref] , [ref] , [ref] , [ref] .
Stochastic six-vertex model in a half-quadrant and half-line open ASEP · Preface · 0.92
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More concretely, we deal with the stochastic six-vertex model (as well as its deformation – the dynamic six–vertex model), that was first introduced in [ref] and whose asymptotic behavior has been recently studied in [ref] , [ref] , [ref] , [ref] , [ref] , [ref] , [ref] .
Stochastic PDE Limit of the Six Vertex Model · Preface · 0.92
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