Taming Reluctant Random Walks In The Positive Quadrant � 2 � , Marni Mishna 2 � , and Yann Ponty 2 � 3 � Jérémie Lumbroso 1 Department of Mathematics � 1 Princeton University Department of Mathematics � 2 Simon Fraser University CNRS Ecole Polytechnique � 3 Inria Saclay June 3 rd , 2016
◆ ❊ ❙ ❲ ✶ ✵ ✵ ✶ ✶ ✵ ✵ ✶ ❩ ✵ ❩ ✵ ❘ ✷ ✶ A lattice model is defined by a set of steps and a region S = { ( ✶ , ✷ ) , ( ✶ , − ✶ ) } ❘ = ❩ × ❩ ≥ ✵
◆ ❊ ❙ ❲ ✶ ✵ ✵ ✶ ✶ ✵ ✵ ✶ ❩ ✵ ❩ ✵ ❘ A lattice model is defined by a set of steps and a region S = { ( ✶ , ✷ ) , ( ✶ , − ✶ ) } ❘ = ❩ × ❩ ≥ ✵ This is a unidimensional model. We could represent it using only { ✷ , − ✶ } .
A lattice model is defined by a set of steps and a region S = { ( ✶ , ✷ ) , ( ✶ , − ✶ ) } ❘ = ❩ × ❩ ≥ ✵ This is a unidimensional model. We could represent it using only { ✷ , − ✶ } . ◆ ❊ ❙ ❲ S = { ( ✶ , ✵ ) , ( ✵ , ✶ ) , ( − ✶ , ✵ ) , ( ✵ , − ✶ ) } ❘ = ❩ ≥ ✵ × ❩ ≥ ✵
A lattice model is defined by a set of steps and a region S = { ( ✶ , ✷ ) , ( ✶ , − ✶ ) } ❘ = ❩ × ❩ ≥ ✵ This is a unidimensional model. We could represent it using only { ✷ , − ✶ } . ◆ ❊ ❙ ❲ S = { ( ✶ , ✵ ) , ( ✵ , ✶ ) , ( − ✶ , ✵ ) , ( ✵ , − ✶ ) } ❘ = ❩ ≥ ✵ × ❩ ≥ ✵ Goal: Efficient uniform random generation of walks in the quarter plane.
Asymptotic Enumeration ❩ ≥ ✵ × ❩ ≥ ✵ ❩ × ❩ ≥ ✵ ❘ ❩ × ❩ plane half-plane quarter-plane S = ✇ ♥ = ✹ ♥ ❤ ♥ ∼ ❝ ✹ ♥ ♥ − ✶ / ✷ q ♥ ∼ ❝ ✹ ♥ ♥ − ✷ / ✸ S = ❤ ♥ ∼ ❝ ✹ . ✹✻ ♥ ♥ − ✸ / ✷ ✇ ♥ = ✺ ♥ q ♥ ∼ ❝ ✹ . ✸ ♥ θ ( ♥ ) Key parameter: Drift( S ) = � ( ✐ , ❥ ) ∈S ( ✐ , ❥ )
Easy case: Zero/Positive drift Consider the following model. ⊳ ✻ ♥ Asymptotically: q ♥ ⊲ The drift is (0,0), naive generation is feasible. A random walk:
Easy case: Zero/Positive drift Consider the following model. ⊳ ✻ ♥ Asymptotically: q ♥ ⊲ The drift is (0,0), naive generation is feasible. A surprisingly efficient strategy is Anticipated rejection . Florentine Algorithm ❬❇❛r❝✉❝❝✐✴P✐♥③❛♥✐✴❙♣r✉❣♥♦❧✐✱ ✾✹✴✾✺❪ Two dimensional analogue ❬❇❛❝❤❡r✴❙♣♦rt✐❡❧❧♦ ✶✹❪ for walks with zero drift The average complexity probably linear.
The difficult case: Reluctant Walks (Negative drift) Consider the following model. ⊳ ✺ . ✵✻ ♥ Asymptotically: q ♥ ⊲ Naive rejection is too inefficient! ( ♣r♦❜ . ≪ ( ✺ . ✵✻ / ✻ ) ♥ ≈ ✶✵ − ✼✹ ) A random walk of a thousand steps.
Summary Positive drift walks are mostly easy to handle
Summary Reluctant (negative drift) walks are less cooperative
Summary Reluctant (negative drift) walks are less cooperative... and require some taming!
Dynamic Programming/Recursive method Theorem ❬❋♦❧❦❧♦r❡❄❪ Random uniform generation of ❦ ❞ -dimensional walks confined to a subset ❘ ⊂ ❩ ❞ is in Θ( ❦ · ♥ + ♥ ❞ + ✶ ) arithmetic operations, using storage for Θ( ♥ ❞ + ✶ ) (large) numbers. Idea: Adapt trivial step-by-step generation into grammar having Θ( ♥ ❞ + ✶ ) NTs � q S ♥ − ✶ ( ① + ✐ , ② + ❥ ) if ♥ > ✵ , q S ( ✐ , ❥ ) ∈S s.t. ♥ ( ① , ② ) = (1) ① + ✐ ≥ ✵ , ② + ❥ ≥ ✵ if ♥ = ✵ ✶ ⇒ Can we do better than Θ( ♥ ✹ ) time?
A better rejection strategy Generate walks in an associated half plane, and wait for a 1/4-plane walk.
Overview of strategy A walk in a 1/4-plane is also a walk in a half plane. 1 ⊆ Convert to a unidimensional walk model. Steps may be in ❘ . 2 ② ≥ − ♠① → Generate unidimensional walks (usually easy) and map to 2D 3 Reject walks that escape the quarter plane (“sub-exponential”?) 4
✺ ✵✻ ♥ q ♥ ❑ ♥ ❤ ♥ ♠ ❑ ♠ The Best 1/2-plane THEOREM ❬❏♦❤♥s♦♥✴▼✐s❤♥❛✴❨❡❛ts ✶✺✰❀●❛r❜✐t✴❘❛s❝❤❡❧ ✶✺✰❪ There is a half plane ② ≥ − ♠① such that (asymptotically) the half plane and quarterplane walks have the same exponential growth factor.
❑ ♥ ❤ ♥ ♠ ❑ ♠ The Best 1/2-plane THEOREM ❬❏♦❤♥s♦♥✴▼✐s❤♥❛✴❨❡❛ts ✶✺✰❀●❛r❜✐t✴❘❛s❝❤❡❧ ✶✺✰❪ There is a half plane ② ≥ − ♠① such that (asymptotically) the half plane and quarterplane walks have the same exponential growth factor. This step set in the quarter plane. ⊳ ✺ . ✵✻ ♥ Asymptotically: q ♥ ⊲
♠ The Best 1/2-plane THEOREM ❬❏♦❤♥s♦♥✴▼✐s❤♥❛✴❨❡❛ts ✶✺✰❀●❛r❜✐t✴❘❛s❝❤❡❧ ✶✺✰❪ There is a half plane ② ≥ − ♠① such that (asymptotically) the half plane and quarterplane walks have the same exponential growth factor. This step set in the quarter plane. ⊳ ✺ . ✵✻ ♥ Asymptotically: q ♥ ⊲ Exponential growth of 1/2-plane walks with these steps in ⊳ ❑ ♥ ) various half-planes. (ie. ❤ ♥ ⊲ ∞ ♠ 0 1/2 3/4 1 2 10 ❑ 5.219 5.075 5.064 5.073 5.156 5.376 5.464
The Best 1/2-plane THEOREM ❬❏♦❤♥s♦♥✴▼✐s❤♥❛✴❨❡❛ts ✶✺✰❀●❛r❜✐t✴❘❛s❝❤❡❧ ✶✺✰❪ There is a half plane ② ≥ − ♠① such that (asymptotically) the half plane and quarterplane walks have the same exponential growth factor. This step set in the quarter plane. ⊳ ✺ . ✵✻ ♥ Asymptotically: q ♥ ⊲ Exponential growth of 1/2-plane walks with these steps in ⊳ ❑ ♥ ) various half-planes. (ie. ❤ ♥ ⊲ ∞ ♠ 0 1/2 3/4 1 2 10 ❑ 5.219 5.075 5.064 5.073 5.156 5.376 5.464 The “best” 1/2-plane is obtained for a slope ♠ = 0.735
Building a grammar for rational walks INPUT: Projected up { ❛ ✐ } , down { ❜ ❥ } and horizontal { ❝ ❦ } steps. Algebraic specification ❬▼❡r❧✐♥✐✴❘♦❣❡rs✴❙♣r✉❣♥♦❧✐✴❱❡rr✐ ✾✾❀ ❉✉❝❤♦♥ ✵✵❪ ❛ , ✐ +¯ ♠✐♥ (¯ ❜ ) � � P = D × P aux ❛ × D + L ❦ × R ❦ − ✐ L ✐ = ❛ ∈ A ❦ = ✐ + ✶ ✇ ( ❛ )= ✐ ❛ , ¯ ♠✐♥ ( ❥ +¯ ❜ ) ❛ ¯ � � � P aux = ε + L ❦ × P aux R ❥ = ❜ × D + L ❦ − ❥ × R ❦ ❦ = ✶ ❜ ∈ A ❦ = ❥ + ✶ ✇ ( ❜ )= − ❥ ❛ , ¯ ♠❛① (¯ ❜ ) � � D = ❝ × D + L ❦ × R ❦ + ε ❝ ∈ ❙ ❦ = ✶ ✇ ( ❝ )= ✵ Number of rules: Θ(( ♠❛① | ❛ ✐ | + ♠❛① | ❜ ❥ | ) ✷ ) – can be huge! Recursive generation ❬❋❧❛❥♦❧❡t✴❩✐♠♠❡r♠❛♥♥✴❱❛♥ ❈✉st❡♠ ✾✹✱ ●♦❧❞✇✉r♠ ✾✺❪ , Boltzmann sampling ❬❉✉❝❤♦♥✴❋❧❛❥♦❧❡t✴▲♦✉❝❤❛r❞✴❙❝❤❛❡✛❡r ✵✹❪ . . . Non-rational slopes... more on this later.
Boltzmann generation success: 18000 steps S = { ( ✶ , ✵ ) , ( ✵ , ✶ ) , ( − ✶ , ✵ ) , ( ✶ , − ✶ ) , ( − ✶ , − ✶ ) , ( − ✷ , − ✶ ) } Using a generic Boltzmann implemention by A. Darrasse
❈ ✶ ❑ ♥ ✸ ✷ ♠ ♥ ♥ r ✶ ✷ ♥ ❑ ♠ ❑ ❈ ✷ ❑ ♥ ♥ r ❬❏♦❤♥s♦♥✴▼✐s❤♥❛✴❨❡❛ts ✶✺✰❪ r Algorithm and its Analysis INPUT: S ⊂ ❩ ✷ (reluctant); Length ♥ OUPUT: Uniform random 1/4-plane walk with steps from S Find optimal slope ♠ (explicit solution computable) 1 If optimal slope is rational, build associated grammar 2 Generate element α from grammar 3 If α confined within quarter plane RETURN α
❬❏♦❤♥s♦♥✴▼✐s❤♥❛✴❨❡❛ts ✶✺✰❪ r Algorithm and its Analysis INPUT: S ⊂ ❩ ✷ (reluctant); Length ♥ OUPUT: Uniform random 1/4-plane walk with steps from S Find optimal slope ♠ (explicit solution computable) 1 If optimal slope is rational, build associated grammar 2 Generate element α from grammar 3 If α confined within quarter plane RETURN α � Time to generate a � Expected time= × Expected # of trials 1/2-plane walk ♠ ♥ − ✸ / ✷ O ( ♥ ) × ❈ ✶ ❑ ♥ = O ( ♥ r − ✶ / ✷ ) since ❑ ♠ = ❑ ❈ ✷ ❑ ♥ ♥ − r
Algorithm and its Analysis INPUT: S ⊂ ❩ ✷ (reluctant); Length ♥ OUPUT: Uniform random 1/4-plane walk with steps from S Find optimal slope ♠ (explicit solution computable) 1 If optimal slope is rational, build associated grammar 2 Generate element α from grammar 3 If α confined within quarter plane RETURN α � Time to generate a � Expected time= × Expected # of trials 1/2-plane walk ♠ ♥ − ✸ / ✷ O ( ♥ ) × ❈ ✶ ❑ ♥ = O ( ♥ r − ✶ / ✷ ) since ❑ ♠ = ❑ ❈ ✷ ❑ ♥ ♥ − r REMARK 1: Optimal slope computable ❬❏♦❤♥s♦♥✴▼✐s❤♥❛✴❨❡❛ts ✶✺✰❪ Bound r ??
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