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<p>Under suitable moment assumptions, we show that a genuinely d-dimensional step-reinforced random walk undergoes a phase transition between recurrence and transience in dimensions <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(d=1,2\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>d</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>, and that it is transient for all reinforcement parameters in dimensions <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(d\ge 3\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>d</mi> <mo>≥</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation>, which solves a conjecture of Bertoin.</p>

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Recurrence-Transience phase transition of the step-reinforced random walk at 1/2

  • 期刊论文

  • 发表日期：

    2025年10月8日

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  • Shuo Qin

摘要

Under suitable moment assumptions, we show that a genuinely d-dimensional step-reinforced random walk undergoes a phase transition between recurrence and transience in dimensions \(d=1,2\) d = 1 , 2 , and that it is transient for all reinforcement parameters in dimensions \(d\ge 3\) d ≥ 3 , which solves a conjecture of Bertoin.