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\(\newcommand {\LWRabsorbtwooptions }[1][]{\LWRabsorboption }\)
\(\def \mathchar {\ifnextchar "\LWRabsorbquotenumber \LWRabsorbnumber }\)
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\(\def \AA {\unicode {x00C5}}\)
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\(\def \O {\unicode {x00D8}}\)
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\(\def \SS {\unicode {x1E9E}}\)
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\( \newcommand {\multicolumn }[3]{#3}\)
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\(\def\upDelta{\unicode{x0394}}\)
\(\def\upEpsilon{\unicode{x0395}}\)
\(\def\upZeta{\unicode{x0396}}\)
\(\def\upEta{\unicode{x0397}}\)
\(\def\upTheta{\unicode{x0398}}\)
\(\def\upVartheta{\unicode{x03F4}}\)
\(\def\upIota{\unicode{x0399}}\)
\(\def\upKappa{\unicode{x039A}}\)
\(\def\upLambda{\unicode{x039B}}\)
\(\def\upMu{\unicode{x039C}}\)
\(\def\upNu{\unicode{x039D}}\)
\(\def\upXi{\unicode{x039E}}\)
\(\def\upOmicron{\unicode{x039F}}\)
\(\def\upPi{\unicode{x03A0}}\)
\(\def\upVarpi{\unicode{x03D6}}\)
\(\def\upRho{\unicode{x03A1}}\)
\(\def\upSigma{\unicode{x03A3}}\)
\(\def\upTau{\unicode{x03A4}}\)
\(\def\upUpsilon{\unicode{x03A5}}\)
\(\def\upPhi{\unicode{x03A6}}\)
\(\def\upChi{\unicode{x03A7}}\)
\(\def\upPsi{\unicode{x03A8}}\)
\(\def\upOmega{\unicode{x03A9}}\)
\(\def\itAlpha{\unicode{x1D6E2}}\)
\(\def\itBeta{\unicode{x1D6E3}}\)
\(\def\itGamma{\unicode{x1D6E4}}\)
\(\def\itDigamma{\mathit{\unicode{x03DC}}}\)
\(\def\itDelta{\unicode{x1D6E5}}\)
\(\def\itEpsilon{\unicode{x1D6E6}}\)
\(\def\itZeta{\unicode{x1D6E7}}\)
\(\def\itEta{\unicode{x1D6E8}}\)
\(\def\itTheta{\unicode{x1D6E9}}\)
\(\def\itVartheta{\unicode{x1D6F3}}\)
\(\def\itIota{\unicode{x1D6EA}}\)
\(\def\itKappa{\unicode{x1D6EB}}\)
\(\def\itLambda{\unicode{x1D6EC}}\)
\(\def\itMu{\unicode{x1D6ED}}\)
\(\def\itNu{\unicode{x1D6EE}}\)
\(\def\itXi{\unicode{x1D6EF}}\)
\(\def\itOmicron{\unicode{x1D6F0}}\)
\(\def\itPi{\unicode{x1D6F1}}\)
\(\def\itRho{\unicode{x1D6F2}}\)
\(\def\itSigma{\unicode{x1D6F4}}\)
\(\def\itTau{\unicode{x1D6F5}}\)
\(\def\itUpsilon{\unicode{x1D6F6}}\)
\(\def\itPhi{\unicode{x1D6F7}}\)
\(\def\itChi{\unicode{x1D6F8}}\)
\(\def\itPsi{\unicode{x1D6F9}}\)
\(\def\itOmega{\unicode{x1D6FA}}\)
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\(\def\upvarbeta{\unicode{x03D0}}\)
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\(\def\updigamma{\unicode{x03DD}}\)
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\(\def\upvarepsilon{\unicode{x03B5}}\)
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\(\def\upeta{\unicode{x03B7}}\)
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\(\def\upvartheta{\unicode{x03D1}}\)
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\(\def\upvarkappa{\unicode{x03F0}}\)
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\(\def\upmu{\unicode{x03BC}}\)
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\(\def\itdigamma{\mathit{\unicode{x03DD}}}\)
\(\def\itdelta{\unicode{x1D6FF}}\)
\(\def\itepsilon{\unicode{x1D716}}\)
\(\def\itvarepsilon{\unicode{x1D700}}\)
\(\def\itzeta{\unicode{x1D701}}\)
\(\def\iteta{\unicode{x1D702}}\)
\(\def\ittheta{\unicode{x1D703}}\)
\(\def\itvartheta{\unicode{x1D717}}\)
\(\def\itiota{\unicode{x1D704}}\)
\(\def\itkappa{\unicode{x1D705}}\)
\(\def\itvarkappa{\unicode{x1D718}}\)
\(\def\itlambda{\unicode{x1D706}}\)
\(\def\itmu{\unicode{x1D707}}\)
\(\def\itnu{\unicode{x1D708}}\)
\(\def\itxi{\unicode{x1D709}}\)
\(\def\itomicron{\unicode{x1D70A}}\)
\(\def\itpi{\unicode{x1D70B}}\)
\(\def\itvarpi{\unicode{x1D71B}}\)
\(\def\itrho{\unicode{x1D70C}}\)
\(\def\itvarrho{\unicode{x1D71A}}\)
\(\def\itsigma{\unicode{x1D70E}}\)
\(\def\itvarsigma{\unicode{x1D70D}}\)
\(\def\ittau{\unicode{x1D70F}}\)
\(\def\itupsilon{\unicode{x1D710}}\)
\(\def\itphi{\unicode{x1D719}}\)
\(\def\itvarphi{\unicode{x1D711}}\)
\(\def\itchi{\unicode{x1D712}}\)
\(\def\itpsi{\unicode{x1D713}}\)
\(\def\itomega{\unicode{x1D714}}\)
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\( \newcommand {\btoa }{\tau ^{>}} \newcommand {\atob }{\tau ^{<}} \newcommand {\iu }{\mathsf {i}} \newcommand {\diff }[1]{\mathsf {d}#1} \newcommand {\PP }{\mathbb {P}} \newcommand {\EE }{\mathbb {E}}
\newcommand {\RR }{\mathbb {R}} \newcommand {\CC }{\mathbb {C}} \newcommand {\NN }{\mathbb {N}} \newcommand {\ZZ }{\mathbb {Z}} \newcommand {\FF }{\mathcal {F}} \newcommand \Aa {\mathcal {A}} \newcommand \Ll
{\mathcal {L}} \newcommand \Dd {\mathcal {D}} \newcommand \ip [2]{\langle #1,#2\rangle } \newcommand \Ind {\mathbb {1}} \newcommand {\Indic }[1]{\Ind _{\{#1\}}} \DeclareMathOperator {\supp }{supp} \newcommand {\ee
}{\mathrm {e}} \DeclareMathOperator {\Exp }{Exp} \)
\( \definecolor {tPrim}{RGB}{0,95,107} \definecolor {TolDarkBlueAcc}{RGB}{43,87,130} \definecolor {TolDarkPink}{RGB}{136,34,85} \newcommand \alert [1]{{\color {tPrim}#1}} \newcommand \Zb {\mathbf {Z}} \newcommand
\dd {\mathrm {d}} \DeclareMathOperator {\oRe }{Re} \DeclareMathOperator {\oIm }{Im} \renewcommand {\Re }{\oRe } \renewcommand {\Im }{\oIm } \)
\(\let \symsf \symsfup \)
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\(\def\Gamma{\unicode{x0393}}\)
\(\def\Digamma{\unicode{x03DC}}\)
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\(\def\Zeta{\unicode{x0396}}\)
\(\def\Eta{\unicode{x0397}}\)
\(\def\Theta{\unicode{x0398}}\)
\(\def\Vartheta{\unicode{x03F4}}\)
\(\def\Iota{\unicode{x0399}}\)
\(\def\Kappa{\unicode{x039A}}\)
\(\def\Lambda{\unicode{x039B}}\)
\(\def\Mu{\unicode{x039C}}\)
\(\def\Nu{\unicode{x039D}}\)
\(\def\Xi{\unicode{x039E}}\)
\(\def\Omicron{\unicode{x039F}}\)
\(\def\Pi{\unicode{x03A0}}\)
\(\def\Varpi{\unicode{x03D6}}\)
\(\def\Rho{\unicode{x03A1}}\)
\(\def\Sigma{\unicode{x03A3}}\)
\(\def\Tau{\unicode{x03A4}}\)
\(\def\Upsilon{\unicode{x03A5}}\)
\(\def\Phi{\unicode{x03A6}}\)
\(\def\Chi{\unicode{x03A7}}\)
\(\def\Psi{\unicode{x03A8}}\)
\(\def\Omega{\unicode{x03A9}}\)
\(\def\alpha{\unicode{x1D6FC}}\)
\(\def\beta{\unicode{x1D6FD}}\)
\(\def\varbeta{\unicode{x03D0}}\)
\(\def\gamma{\unicode{x1D6FE}}\)
\(\def\digamma{\mathit{\unicode{x03DD}}}\)
\(\def\delta{\unicode{x1D6FF}}\)
\(\def\epsilon{\unicode{x1D716}}\)
\(\def\varepsilon{\unicode{x1D700}}\)
\(\def\zeta{\unicode{x1D701}}\)
\(\def\eta{\unicode{x1D702}}\)
\(\def\theta{\unicode{x1D703}}\)
\(\def\vartheta{\unicode{x1D717}}\)
\(\def\iota{\unicode{x1D704}}\)
\(\def\kappa{\unicode{x1D705}}\)
\(\def\varkappa{\unicode{x1D718}}\)
\(\def\lambda{\unicode{x1D706}}\)
\(\def\mu{\unicode{x1D707}}\)
\(\def\nu{\unicode{x1D708}}\)
\(\def\xi{\unicode{x1D709}}\)
\(\def\omicron{\unicode{x1D70A}}\)
\(\def\pi{\unicode{x1D70B}}\)
\(\def\varpi{\unicode{x1D71B}}\)
\(\def\rho{\unicode{x1D70C}}\)
\(\def\varrho{\unicode{x1D71A}}\)
\(\def\sigma{\unicode{x1D70E}}\)
\(\def\varsigma{\unicode{x1D70D}}\)
\(\def\tau{\unicode{x1D70F}}\)
\(\def\upsilon{\unicode{x1D710}}\)
\(\def\phi{\unicode{x1D719}}\)
\(\def\varphi{\unicode{x1D711}}\)
\(\def\chi{\unicode{x1D712}}\)
\(\def\psi{\unicode{x1D713}}\)
\(\def\omega{\unicode{x1D714}}\)
A growth-fragmentation found in the cone excursions of Brownian motion (and in the quantum disc)
Cone excursions
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• Take a planar Brownian motion, consider two types of ‘cone excursion’:
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• ‘Cone-free times’ (between boundary-to-apex excursions) form a regenerative set
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• The path is cut into cone excursions between said times
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• Write \(\btoa \) for boundary-to-apex inverse local time
<3->
Targeting a time
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• Take a point on a Brownian path \(B\) (at time \(t\)), and progressively...
-
• ...cut out boundary-to-apex cone excursions not containing that time...
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• ...chop out extra path sections at the end, keeping the path inside a cone...
-
• ...and so on.
<1>
<2>
<3>
<4>
<5>
<6>
<7>
<8>
<9>
<10>
<11>
<12>
<13>
Targeting a time (more precisely)
-
• Fix \(t>0\), write \(\btoa _t\) for boundary-to-apex inverse local time of \((B_s : s\le t)\)
-
• Then fix \(a\ge 0\), and...
-
• ...let \(\rho _t(a)\) be the smallest time making \(B[\btoa _t(a), \rho _t(a)]\) a whole-path cone excursion
-
• ...and let \(e^a_t(s) = B_{s + \btoa _t(a)} - B_{\rho _t(a)}\).
-
• Call \(e^a_t\) the excursion targeting \(t\) (at level \(a\))
Targeting multiple times
-
• We have the excursion targeting \(t\)
-
• What if we target some other \(t'\)?
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• Every piece cut out while targeting \(t\) is one which is included in the excursion targeting some other \(t'\)
-
• Consider targeting every time simultaneously
-
• There is some kind of branching process for us to capture
Summarising the path targeting \(t\)
-
• Map the cone with apex angle \(\theta \) to the positive quadrant \(\RR _+^2\); standard Brownian motion becomes correlated
-
• The initial displacement of the excursion targeting \(t\) at local time \(a\): \(e_t^a(0) \in \RR _+^2\)
-
• In the case \(\theta =2\pi /3\) look at its \(\ell ^1\)-norm: \(Z_t(a) = \lVert e_t^a(0) \rVert _1\)
Growth-fragmentations
A growth-fragmentation is:
-
• a system of particles (excursions targeting each time \(t\))...
-
• ...each summarised by a trait (\(\ell ^1\)-norm of its initial displacement)...
-
• ...each of which is a Markov process when viewed on its own...
-
• ...whose path only jumps down...
-
• ...and each jump of which is accompanied by the birth of another particle, conditionally independent given initial trait value
Main result
We do all this starting with \(B\) given by a boundary-to-apex excursion with fixed initial value \(B_0 = z \in \RR ^2_+\).
Relationship with the quantum disc
-
• The quantum disc is a ball of radius \(1\) in the complex plane, loosely speaking endowed with a Riemannian metric \(e^{\gamma h(z)}(dx^2 + dy^2)\) at \(z=x+iy\), where \(h\) is a Gaussian free field
-
• Take \(\gamma =\sqrt {8/3}\) and mark the point \(-i\) on the boundary
-
• Draw counterclockwise space-filling SLE\(_{6}\) curve started at \(-i\), targeting the same point
-
• As the curve fills space targeting a particular point, it cuts out ‘bubbles’ not containing the point (which are explored by another branch)
-
• Our growth-fragmentation describes the total (left and right) boundary length of the branches
Prior art
-
• Boundary-to-apex cone excursions: described (in time reversal) by Duplantier, Miller and Sheffield (2021)
-
• Whole-path cone excursions: described by Le Gall (1987)
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• Another growth-fragmentation in planar excursions: Aïdékon and Da Silva (2022)
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• Relationship with the quantum disc: the ‘mating of trees’, Duplantier, Miller and Sheffield (2021) and Ang and Gwynne (2021)
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• Growth-fragmentation in the ‘conformal percolation interface of the conformal loop ensemble carpet’: Miller, Sheffield and Werner (2020); our work in some sense lies at the boundary of their parameter regime
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• Growth-fragmentation in the Brownian disc (via the snake construction): Le Gall and Riera (2020)
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• Growth-fragmentation from random planar maps: Bertoin, (Budd,) Curien and Kortchemski (2018)
Describing the growth-fragmentation
-
• Take a uniform time \(T \in (0,\zeta )\)
-
• Consider the excursion targeting \(T\), corresponding to the process \((Z_T(a) : 0\le a \le \zeta _T)\)
-
• Time-reverse it to get \(S(a) = Z_T((\zeta _T-a)^-)\), \(0\le a \le \zeta _T\)
A beautiful construction of a conditioned \(3/2\)-stable process
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• Everything here is in reverse time!
-
• Let \(W,W'\) be independent (correlated) Brownian motions, \(X(t) = W'(t)\) for \(t\ge 0\) and \(X(t) = W(-t)\) for \(t\le 0\)
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• Write \(\atob \) for the inverse local time of apex-to-boundary excursions of \(W'\)
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• Let \(\sigma (a)\) be the first time (for \(W\)) the displaced quadrant \(W(\sigma (a)) + \RR ^2_+\) contains \(X[-\sigma (a),\atob (a)]\).
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• Write \(Y(a) = X(\atob (a)) - X(-\sigma (a))\) and \(\mathcal {S}(a) = \lVert Y(a) \rVert _1\)
<1-4>
<5->
Sketch of the proof
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• Blue jumps arise from seeing new apex-to-boundary excursions (in \(W'\))
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• They form the jumps of a \(3/2\)-stable process which jumps directly up or right (DMS ’21)
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• Red jumps arise from seeing new whole-path excursions (in \(X\))
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• These form the jumps of a \(1/2\)-stable process of unknown jump distribution (LG ’87)
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• To complete the proof:
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– Find the distribution of red jumps
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– Show \(\mathcal {S}\) is Markov
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– Show that red jumps occur at rate \(1/\mathcal {S}(a)\)
Relationship with the growth-fragmentation
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• Let \(\bar {e}^a(b) = X((\atob (a)-b)^-)\) for \(0\le b\le \atob (a)+\sigma (a)\) (time-reversal of stopped \(X\))
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• Let \(A\) be ‘distributed’ according to Lebesgue measure on \((0,\infty )\)
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• Then \((\bar {e}^A, A)\) has the same distribution as a generic whole-path excursion together with a time uniformly chosen within its lifetime
Summary
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• Isolating excursions targeting a given time (by cutting out excursions targeting others) gives rise to a growth-fragmentation with law connected to a stable process
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• Along the way we found:
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• We also obtain the law of the ‘locally largest’ particle (whose trait changes the least at each branch point)...
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• ...and find a special martingale whose limit law is that of the lifetime of a typical excursion (recovering a result about the volume of Boltzmann triangulations)
Posterior art
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• The growth-fragmentation is the same one found in the Brownian disc (Le Gall and Riera) and Boltzmann triangulations (Bertoin, Curien and Kortchemski)
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• It is in the same class as the one found in percolation of CLE carpets (Miller, Sheffield and Werner) and metric exploration of random planar maps (Bertoin, Budd, Curien and Kortchemski)
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• It may be possible to derive our results with quantum gravity arguments, but we use nothing but an analysis of Brownian motion
Open questions
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• Is there a typed version, to distinguish whether a ‘particle’ corresponds to a boundary-to-apex or interior-to-apex (whole-path) excursion?
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• What about \(\theta \ne 2\pi /3\)? (cf. half plane excursions, planar maps, CLE carpet)
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• Connected to that: are there other nice constructions of conditioned stable processes that should appear?
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• Can we start things at zero?
