We study a pair of independent searchers competing for a target under restarts and find that introduction of restarts tends to enhance the search efficiency of an already efficient searcher. As a result, the difference between the search probabilities of the individual searchers increases when the system is subject to restarts. This result holds true independent of the identity of individual searchers or the specific details of the distribution of restart times. However, when only one of a pair of searchers is subject to restarts while the other evolves in an unperturbed manner, a concept termed as subsystem restarts, we find that the search probability exhibits a nonmonotonic dependence on the restart rate. We also study the mean search time for a pair of run and tumble and Brownian searchers when only the run and tumble particle is subject to restarts. We find that, analogous to restarting the whole system, the mean search time exhibits a nonmonotonic dependence on restart rates.
REFERENCES
1.
S.
Gupta
and A. M.
Jayannavar
, “Stochastic resetting: A (very) brief review
,” Front. Phys.
10
, 789097
(2022
). 2.
R. K.
Singh
, K.
Gorska
, and T.
Sandev
, “General approach to stochastic resetting
,” Phys. Rev. E
105
, 064133
(2022
). 3.
M. R.
Evans
and S. N.
Majumdar
, “Diffusion with stochastic resetting
,” Phys. Rev. Lett.
106
, 160601
(2011
). 4.
R. K.
Singh
, T.
Sandev
, A.
Iomin
, and R.
Metzler
, “Backbone diffusion and first-passage dynamics in a comb structure with confining branches under stochastic resetting
,” J. Phys. A: Math. Theor.
54
, 404006
(2021
). 5.
M.
Luby
, A.
Sinclair
, and D.
Zuckerman
, “Optimal speedup of Las Vegas algorithms
,” Inf. Process. Lett.
47
, 173
–180
(1993
). 6.
S.
Reuveni
, “Optimal stochastic restart renders fluctuations in first passage times universal
,” Phys. Rev. Lett.
116
, 170601
(2016
). 7.
A.
Pal
and S.
Reuveni
, “First passage under restart
,” Phys. Rev. Lett.
118
, 030603
(2017
). 8.
M. R.
Evans
and S. N.
Majumdar
, “Diffusion with optimal resetting
,” J. Phys. A: Math. Theor.
44
, 435001
(2011
). 9.
I.
Eliazar
and S.
Reuveni
, “Mean-performance of sharp restart I: Statistical roadmap
,” J. Phys. A: Math. Theor.
53
, 405004
(2020
). 10.
I.
Eliazar
and S.
Reuveni
, “Mean-performance of sharp restart II: Inequality roadmap
,” J. Phys. A: Math. Theor.
54
, 355001
(2021
). 11.
A.
Pal
, A.
Kundu
, and M. R.
Evans
, “Diffusion under time-dependent resetting
,” J. Phys. A: Math. Theor.
49
, 225001
(2016
). 12.
U.
Bhat
, C.
De Bacco
, and S.
Redner
, “Stochastic search with Poisson and deterministic resetting
,” J. Stat. Mech.: Theor. Exp.
2016
, 083401
. 13.
M. R.
Evans
and S. N.
Majumdar
, “Run and tumble particle under resetting: A renewal approach
,” J. Phys. A: Math. Theor.
51
, 475003
(2018
). 14.
J.
Masoliver
, “Telegraphic processes with stochastic resetting
,” Phys. Rev. E
99
, 012121
(2019
). 15.
A.
Nagar
and S.
Gupta
, “Diffusion with stochastic resetting at power-law times
,” Phys. Rev. E
93
, 060102
(2016
). 16.
V.
Domazetoski
, A.
Masó-Puigdellosas
, T.
Sandev
, V.
Méndez
, A.
Iomin
, and L.
Kocarev
, “Stochastic resetting on comblike structures
,” Phys. Rev. Res.
2
, 033027
(2020
). 17.
M. R.
Evans
and S. N.
Majumdar
, “Diffusion with resetting in arbitrary spatial dimension
,” J. Phys. A: Math. Theor.
47
, 285001
(2014
). 18.
S.
Ahmad
, I.
Nayak
, A.
Bansal
, A.
Nandi
, and D.
Das
, “First passage of a particle in a potential under stochastic resetting: A vanishing transition of optimal resetting rate
,” Phys. Rev. E
99
, 022130
(2019
). 19.
S.
Ahmad
, K.
Rijal
, and D.
Das
, “First passage in the presence of stochastic resetting and a potential barrier
,” Phys. Rev. E
105
, 044134
(2022
). 20.
A.
Pal
and V.
Prasad
, “First passage under stochastic resetting in an interval
,” Phys. Rev. E
99
, 032123
(2019
). 21.
A.
Chechkin
and I. M.
Sokolov
, “Random search with resetting: A unified renewal approach
,” Phys. Rev. Lett.
121
, 050601
(2018
). 22.
M. R.
Evans
, S. N.
Majumdar
, and G.
Schehr
, “An exactly solvable predator prey model with resetting
,” J. Phys. A: Math. Theor.
55
, 274005
(2022
). 23.
S.
Belan
, “Restart could optimize the probability of success in a Bernoulli trial
,” Phys. Rev. Lett.
120
, 080601
(2018
). 24.
R. K.
Singh
and S.
Singh
, “Capture of a diffusing lamb by a diffusing lion when both return home
,” Phys. Rev. E
106
, 064118
(2022
). 25.
R. K.
Singh
, T.
Sandev
, and S.
Singh
, “Bernoulli trial under restarts: A comparative study of resetting transitions
,” Phys. Rev. E
108
, L052106
(2023
). 26.
O.
Vilk
, M.
Assaf
, and B.
Meerson
, “Fluctuations and first-passage properties of systems of Brownian particles with reset
,” Phys. Rev. E
106
, 024117
(2022
). 27.
A.
Nagar
and S.
Gupta
, “Stochastic resetting in interacting particle systems: A review
,” J. Phys. A: Math. Theor.
56
, 283001
(2023
). 28.
M. R.
Evans
, S. N.
Majumdar
, and G.
Schehr
, “Stochastic resetting and applications
,” J. Phys. A: Math. Theor.
53
, 193001
(2020
). 29.
F.
Bartumeus
, J.
Catalan
, U. L.
Fulco
, M. L.
Lyra
, and G. M.
Viswanathan
, “Optimizing the encounter rate in biological interactions: Lévy versus Brownian strategies
,” Phys. Rev. Lett.
88
, 097901
(2002
). 30.
A.
James
, M. J.
Plank
, and R.
Brown
, “Optimizing the encounter rate in biological interactions: Ballistic versus Lévy versus Brownian strategies
,” Phys. Rev. E
78
, 051128
(2008
). 31.
V. V.
Palyulin
, A. V.
Chechkin
, and R.
Metzler
, “Lévy flights do not always optimize random blind search for sparse targets
,” Proc. Natl. Acad. Sci. U.S.A.
111
, 2931
–2936
(2014
). 32.
V. V.
Palyulin
, G.
Blackburn
, M. A.
Lomholt
, N. W.
Watkins
, R.
Metzler
, R.
Klages
, and A. V.
Chechkin
, “First passage and first hitting times of Lévy flights and Lévy walks
,” New J. Phys.
21
, 103028
(2019
). 33.
V. V.
Palyulin
, A. V.
Chechkin
, R.
Klages
, and R.
Metzler
, “Search reliability and search efficiency of combined Lévy–Brownian motion: Long relocations mingled with thorough local exploration
,” J. Phys. A: Math. Theor.
49
, 394002
(2016
). 34.
E. P.
Raposo
, S. V.
Buldyrev
, M. G. E.
Da Luz
, M. C.
Santos
, H. E.
Stanley
, and G. M.
Viswanathan
, “Dynamical robustness of Lévy search strategies
,” Phys. Rev. Lett.
91
, 240601
(2003
). 35.
O.
Bénichou
, C.
Loverdo
, M.
Moreau
, and R.
Voituriez
, “Intermittent search strategies
,” Rev. Mod. Phys.
83
, 81
(2011
).36.
M. A.
Lomholt
, K.
Tal
, R.
Metzler
, and K.
Joseph
, “Lévy strategies in intermittent search processes are advantageous
,” Proc. Natl. Acad. Sci. U.S.A.
105
, 11055
–11059
(2008
). 37.
M. A.
Lomholt
, T.
Ambjörnsson
, and R.
Metzler
, “Optimal target search on a fast-folding polymer chain with volume exchange
,” Phys. Rev. Lett.
95
, 260603
(2005
). 38.
O.
Vilk
, Y.
Orchan
, M.
Charter
, N.
Ganot
, S.
Toledo
, R.
Nathan
, and M.
Assaf
, “Ergodicity breaking in area-restricted search of avian predators
,” Phys. Rev. X
12
, 031005
(2022
).39.
H. S.
Carslaw
, Introduction to the Mathematical Theory of the Conduction of Heat in Solids
(Macmillan and Company Ltd.
, 1906
).40.
E.
Abad
, S. B.
Yuste
, and K.
Lindenberg
, “Survival probability of an immobile target in a sea of evanescent diffusive or subdiffusive traps: A fractional equation approach
,” Phys. Rev. E
86
, 061120
(2012
). 41.
S.
Redner
, A Guide to First-Passage Processes
(Cambridge University Press
, 2001
).42.
R.
Metzler
, S.
Redner
, and G.
Oshanin
, First-Passage Phenomena and Their Applications
(World Scientific
, 2014
), Vol. 35.43.
K.
Malakar
, V.
Jemseena
, A.
Kundu
, K. V.
Kumar
, S.
Sabhapandit
, S. N.
Majumdar
, S.
Redner
, and A.
Dhar
, “Steady state, relaxation and first-passage properties of a run-and-tumble particle in one-dimension
,” J. Stat. Mech.: Theor. Exp.
2018
, 043215
. 44.
R.
Majumder
, R.
Chattopadhyay
, and S.
Gupta
, “Kuramoto model subject to subsystem resetting: How resetting a part of the system may synchronize the whole of it
,” Phys. Rev. E
109
, 064137
(2024
). 45.
G. H.
Weiss
, K. E.
Shuler
, and K.
Lindenberg
, “Order statistics for first passage times in diffusion processes
,” J. Stat. Phys.
31
, 255
–278
(1983
). 46.
S. B.
Yuste
and K.
Lindenberg
, “Order statistics for first passage times in one-dimensional diffusion processes
,” J. Stat. Phys.
85
, 501
–512
(1996
). 47.
S. B.
Yuste
and L.
Acedo
, “Order statistics of the trapping problem
,” Phys. Rev. E
64
, 061107
(2001
). 48.
D.
Hartich
and A.
Godec
, “Duality between relaxation and first passage in reversible Markov dynamics: Rugged energy landscapes disentangled
,” New J. Phys.
20
, 112002
(2018
). 49.
S. D.
Lawley
and J. B.
Madrid
, “A probabilistic approach to extreme statistics of Brownian escape times in dimensions 1, 2, and 3
,” J. Nonlinear Sci.
30
, 1207
–1227
(2020
). 50.
D. S.
Grebenkov
, R.
Metzler
, and G.
Oshanin
, “From single-particle stochastic kinetics to macroscopic reaction rates: Fastest first-passage time of
random walkers
,” New J. Phys.
22
, 103004
(2020
). 51.
S.
Linn
and S. D.
Lawley
, “Extreme hitting probabilities for diffusion
,” J. Phys. A: Math. Theor.
55
, 345002
(2022
). © 2025 Author(s). Published under an exclusive license by AIP Publishing.
2025
Author(s)
You do not currently have access to this content.
