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Conformally Invariant Fields Out of Brownian Loop Soups

A. Jego Titus Lupu Wei Qian
Aug 2026 · Memoirs of the American Mathematical Society · 0 citations · 73 references

Abstract

<p> Consider a Brownian loop soup <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper L Subscript upper D Superscript theta"> <mml:semantics> <mml:msubsup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">L</mml:mi> </mml:mrow> <mml:mi>D</mml:mi> <mml:mi> θ </mml:mi> </mml:msubsup> <mml:annotation encoding="application/x-tex">\mathcal {L}_D^\theta</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with subcritical intensity <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="theta element-of left-parenthesis 0 comma 1 slash 2 right-bracket"> <mml:semantics> <mml:mrow> <mml:mi> θ </mml:mi> <mml:mo> ∈ </mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> <mml:mo stretchy="false">]</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\theta \in (0,1/2]</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in some 2D bounded simply connected domain <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper D"> <mml:semantics> <mml:mi>D</mml:mi> <mml:annotation encoding="application/x-tex">D</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . We define and study the properties of a conformally invariant field <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="h Subscript theta"> <mml:semantics> <mml:msub> <mml:mi>h</mml:mi> <mml:mi> θ </mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">h_\theta</mml:annotation> </mml:semantics> </mml:math> </inline-formula> naturally associated to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper L Subscript upper D Superscript theta"> <mml:semantics> <mml:msubsup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">L</mml:mi> </mml:mrow> <mml:mi>D</mml:mi> <mml:mi> θ </mml:mi> </mml:msubsup> <mml:annotation encoding="application/x-tex">\mathcal {L}_D^\theta</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . Informally, this field is a signed version of the local time of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper L Subscript upper D Superscript theta"> <mml:semantics> <mml:msubsup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">L</mml:mi> </mml:mrow> <mml:mi>D</mml:mi> <mml:mi> θ </mml:mi> </mml:msubsup> <mml:annotation encoding="application/x-tex">\mathcal {L}_D^\theta</mml:annotation> </mml:semantics> </mml:math> </inline-formula> to the power <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="1 minus theta"> <mml:semantics> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo> − </mml:mo> <mml:mi> θ </mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">1-\theta</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . When <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="theta equals 1 slash 2"> <mml:semantics> <mml:mrow> <mml:mi> θ </mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">\theta = 1/2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="h Subscript theta"> <mml:semantics> <mml:msub> <mml:mi>h</mml:mi> <mml:mi> θ </mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">h_\theta</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a Gaussian free field (GFF) in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper D"> <mml:semantics> <mml:mi>D</mml:mi> <mml:annotation encoding="application/x-tex">D</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . </p> <p> Our construction of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="h Subscript theta"> <mml:semantics> <mml:msub> <mml:mi>h</mml:mi> <mml:mi> θ </mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">h_\theta</mml:annotation> </mml:semantics> </mml:math> </inline-formula> relies on the multiplicative chaos <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper M Subscript gamma"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">M</mml:mi> </mml:mrow> <mml:mi>

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