<p>This paper is concerned with a natural variant of the contact process modeling the spread of knowledge on the integer lattice. Each site is characterized by its knowledge, measured by a real number ranging from 0 = ignorant to 1 = omniscient. Neighbors interact at rate <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\lambda $</EquationSource> <EquationSource Format="MATHML"><math> <mi>λ</mi> </math></EquationSource> </InlineEquation>, which results in both neighbors attempting to teach each other a fraction <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\mu $</EquationSource> <EquationSource Format="MATHML"><math> <mi>μ</mi> </math></EquationSource> </InlineEquation> of their knowledge, and individuals die at rate one, which results in a new individual with no knowledge. Starting with a single omniscient site, our objective is to study whether the total amount of knowledge on the lattice converges to zero (extinction) or remains bounded away from zero (survival). The process dies out when <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\lambda \le \lambda _c$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>λ</mi> <mo>≤</mo> <msub> <mi>λ</mi> <mi>c</mi> </msub> </mrow> </math></EquationSource> </InlineEquation> and/or <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\mu = 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>μ</mi> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\lambda _c$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>λ</mi> <mi>c</mi> </msub> </math></EquationSource> </InlineEquation> denotes the critical value of the contact process. In contrast, we prove that, for all <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\lambda > \lambda _c$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>λ</mi> <mo>></mo> <msub> <mi>λ</mi> <mi>c</mi> </msub> </mrow> </math></EquationSource> </InlineEquation>, there is a unique phase transition in the direction of <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\mu $</EquationSource> <EquationSource Format="MATHML"><math> <mi>μ</mi> </math></EquationSource> </InlineEquation>, and for all <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\mu > 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>μ</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, there is a unique phase transition in the direction of <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\lambda $</EquationSource> <EquationSource Format="MATHML"><math> <mi>λ</mi> </math></EquationSource> </InlineEquation>. Our proof of survival relies on block constructions showing more generally convergence of the knowledge to infinity, while our proof of extinction relies on martingale techniques showing more generally an exponential decay of the knowledge.</p>

Contact Process for the Spread of Knowledge

期刊论文
发表日期：

2026年1月20日

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Nicolas Lanchier,
Max Mercer,
Hyunsik Yun

摘要

This paper is concerned with a natural variant of the contact process modeling the spread of knowledge on the integer lattice. Each site is characterized by its knowledge, measured by a real number ranging from 0 = ignorant to 1 = omniscient. Neighbors interact at rate $\lambda $ $λ$ , which results in both neighbors attempting to teach each other a fraction $\mu $ $μ$ of their knowledge, and individuals die at rate one, which results in a new individual with no knowledge. Starting with a single omniscient site, our objective is to study whether the total amount of knowledge on the lattice converges to zero (extinction) or remains bounded away from zero (survival). The process dies out when $\lambda \le \lambda _c$ $λ \leq λ_{c}$ and/or $\mu = 0$ $μ = 0$ , where $\lambda _c$ $λ_{c}$ denotes the critical value of the contact process. In contrast, we prove that, for all $\lambda > \lambda _c$ $λ > λ_{c}$ , there is a unique phase transition in the direction of $\mu $ $μ$ , and for all $\mu > 0$ $μ > 0$ , there is a unique phase transition in the direction of $\lambda $ $λ$ . Our proof of survival relies on block constructions showing more generally convergence of the knowledge to infinity, while our proof of extinction relies on martingale techniques showing more generally an exponential decay of the knowledge.