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<p>We prove that for Bernoulli percolation on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper Z Superscript d"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">Z</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> <mml:annotation encoding="application/x-tex">\mathbb {Z}^d</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="d greater-than-or-equal-to 2"> <mml:semantics> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>≥<!-- ≥ --></mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">d\geq 2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, the percolation density is an analytic function of the parameter in the supercritical interval. For this we introduce some techniques that have further implications. In particular, we prove that the susceptibility is analytic in the subcritical interval for all transitive short- or long-range models, and that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p Subscript c Superscript b o n d Baseline greater-than 1 slash 2"> <mml:semantics> <mml:mrow> <mml:msubsup> <mml:mi>p</mml:mi> <mml:mi>c</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>b</mml:mi> <mml:mi>o</mml:mi> <mml:mi>n</mml:mi> <mml:mi>d</mml:mi> </mml:mrow> </mml:msubsup> <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">p_c^{bond} >1/2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for certain families of triangulations for which Benjamini & Schramm conjectured that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p Subscript c Superscript s i t e Baseline less-than-or-equal-to 1 slash 2"> <mml:semantics> <mml:mrow> <mml:msubsup> <mml:mi>p</mml:mi> <mml:mi>c</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>s</mml:mi> <mml:mi>i</mml:mi> <mml:mi>t</mml:mi> <mml:mi>e</mml:mi> </mml:mrow> </mml:msubsup> <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">p_c^{site} \leq 1/2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>.</p>
TL;DRIt is proved that the susceptibility is analytic in the subcritical interval for all transitive short- or long-range models, and that the percolation density is an analytic function of the parameter in the supercritical interval. (Semantic Scholar)
