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### Large deviations [article]

Satya N. Majumdar, Gregory Schehr
2017 arXiv   pre-print
For large N , we can use Stirling's approximation N ! ∼ √ 2πN e N ln NN to write P (M = c N, N ) = 1 2 N N ! (c N )!((1 − c) N )!  ...  − lim N →∞ 1 β N ln Z.  ...

### Persistence in Nonequilibrium Systems [article]

Satya N. Majumdar
1999 arXiv   pre-print
The idea is to consider the generating function, P (p, t) = ∞ n=0 p n P n (t) (8) where P n (t) is the probability of n zero crossings in time t of the "effective" single site process.  ...  Since the "clipped" process σ(T ) can take values ±1 only, one can express A(T ) as, A(T ) = ∞ n=0 (−1) n P n (T ), (4) where P n (T ) is the probability that the interval T contains n zeros of φ(T ).  ...

### Ising model with stochastic resetting [article]

Matteo Magoni, Satya N. Majumdar, Gregory Schehr
2020 arXiv   pre-print
This property makes the 1d Glauber dynamics exactly solvable as the evolution equation for the n-point correlation functions only involve n-point functions, i.e. it satisfies a closure property [53] .  ...  Note that under the Glauber dynamics, the magnetisation m(t) = (1/N ) i s i (t) evolves deterministically with time t.  ...

### Brownian Functionals in Physics and Computer Science [article]

Satya N. Majumdar
2005 arXiv   pre-print
Langevin's formulation in Eq. (10) also makes a correspondence between Brownian motion and the random walk problem where the position x n of a random walker after n steps evolves via x n = x n−1 + ξ n  ...  Examples of rooted planar trees with n + 1 = 4 vertices are shown in Fig. (3) . There are in general C n+1 = 1 n+1 2n n number of possible rooted planar tree configurations with (n + 1) vertices.  ...

### Real-space Condensation in Stochastic Mass Transport Models [article]

Satya N. Majumdar
2009 arXiv   pre-print
In the thermodynamic limit N → ∞, V → ∞ but with the density ρ = N/V fixed, as one reduces the temperature below a certain critical value T c (ρ) in d > 2, macroscopically large number of particles (∝  ...  Consider an ideal gas of N bosons in a d-dimensional hybercubic box of volume V = L d .  ...

### Optimal Resetting Brownian Bridges [article]

Benjamin De Bruyne, Satya N. Majumdar, Gregory Schehr
2022 arXiv   pre-print
Optimal Resetting Brownian Bridges: Supplemental Material Benjamin De Bruyne, 1 Satya N.  ...  Perhaps, the effect of resetting is best seen in the simple model of diffusion introduced by Evans and Majumdar [41] .  ...

### Stochastic Resetting and Applications [article]

Martin R. Evans, Satya N. Majumdar, Gregory Schehr
2020 arXiv   pre-print
It is useful to consider the distance y of the walker at time n from the maximum position upto time n: y(n) = m(n)−x(n).  ...  This results in the solution of (3.59) p * (x) = p 0 + ∞ n=1 2r N L(r − D n ) cos(nπx/L) N i=1 cos(nπX ri /L) (3.62) where p 0 is chosen to normalise the probability.  ...

### Stability of large complex systems with heterogeneous relaxation dynamics [article]

Pierre Mergny, Satya N. Majumdar
2021 arXiv   pre-print
For a specific flat configuration a_i = 1 + σi-1/N, we obtain an explicit parametric solution for the limiting (as N→∞) spectral density for arbitrary T and σ.  ...  The a_i>0's are the intrinsic decay rates, J_ij is a real symmetric (N× N) Gaussian random matrix and √(T) measures the strength of pairwise interaction between different species.  ...  ,b N ) ∝ e − N 2T N i=1b 2 i ∆(b 1 , . . .b N ) 2 U(N ) e N T Tr (A U Diag(b1,...  ...

### Extreme Eigenvalues of Wishart Matrices: Application to Entangled Bipartite System [article]

Satya N. Majumdar
2010 arXiv   pre-print
constant A N is given by P N (x) = A N x −N/2 (1−N x) (N 2 +N −4)/2 2 F 1 N + 2 2 , N − 1 2 , N 2 + N − 2 2 , − 1 − N x x (1.4.15) where 2 F A N = N Γ(N ) Γ(N 2 /2) 2 N −1 Γ(N/2) Γ((N 2 + N − 2)/2) . (  ...  N (1/N − ǫ) = 1/N 1/N −ǫ P N (x) dx where ǫ << 1/N .  ...

### Freezing transitions of Brownian particles in confining potentials [article]

Gabriel Mercado-Vásquez, Denis Boyer, Satya N. Majumdar
2022 arXiv   pre-print
We focus on the cases in which the external potential is confining, of the form v(x)=k|x-x_0|^n/n, and where the particle's initial position coincides with x_0.  ...  The phase diagram in the (x_0,n)-plane then exhibits three dynamical phases and metastability, with a "triple" point at (x_0/c≃ 0.17185, n≃ 0.39539).  ...  0 ) n ) − (n − 2) (1 − x 0 ) n (n + 1) 2 (n + 2) − 2x 0 [(n + 2 − 4x 0 )x n 0 + (n − 2 + 4x 0 ) (1 − x 0 ) n ] (n + 1) 2 (n + 2) . ( 37 ) Setting the above expression to zero and combining it with Eq.  ...

### Clustering of advected passive sliders on a fluctuating surface [article]

Apoorva Nagar, Mustansir Barma, Satya N. Majumdar
2004 arXiv   pre-print
Satya N. Majumdar is at Laboratoire de Physique Quantique, Universit'e Paul Sabatier, 31062 Toulouse Cedex, France. E-mail:  ...  They actualy evaluated the n-point correlation function.  ...

### Fluctuation-dominated phase ordering at a mixed order transition [article]

Mustansir Barma, Satya N. Majumdar, David Mukamel
2019 arXiv   pre-print
n=0 p n (r) = 1 which gives ∞ n=0p n (s) = 1/s. Using the results forp n (s) for n ≥ 1 in Eq.  ...  Q(l 1 ) P (l 2 ) P (l 3 ) · · · P (l n ) Q(l n+1 ) δ(l 1 + l 2 + · · · + l n + l n+1 − r) .  ...

### A note on limit shapes of minimal difference partitions [article]

Alain Comtet, Satya N. Majumdar, Sanjib Sabhapandit
2008 arXiv   pre-print
), (2) where − → n ≡ (n 1 , n 2 ) = (b, a)/ √ a 2 + b 2 is the unit vector orthogonal to − − → P Q and φ( − → n ) = −n 1 ln n 1 n 1 + n 2 − n 2 ln n 2 n 1 + n 2 . (3) Heuristically one expects that in  ...  = {n i } δ E − ∞ i=1 n i ǫ i δ N − ∞ i=1 n i . (18) For both unrestricted and restricted partitions, one can readily check that the grand partition function Z(e −β , z) = N E z N e −βE ρ(E, N ), in the  ...

### Intermittent resetting potentials [article]

Gabriel Mercado-Vásquez, Denis Boyer, Satya N. Majumdar, Grégory Schehr
2020 arXiv   pre-print
We study the non-equilibrium steady states and first passage properties of a Brownian particle with position X subject to an external confining potential of the form V(X)=μ|X|, and that is switched on and off stochastically. Applying the potential intermittently generates a physically realistic diffusion process with stochastic resetting toward the origin, a topic which has recently attracted a considerable interest in a variety of theoretical contexts but has remained challenging to implement
more » ... n lab experiments. The present system exhibits rich features, not observed in previous resetting models. The mean time needed by a particle starting from the potential minimum to reach an absorbing target located at a certain distance can be minimized with respect to the switch-on and switch-off rates. The optimal rates undergo continuous or discontinuous transitions as the potential stiffness μ is varied across non-trivial values. A discontinuous transition with metastable behavior is also observed for the optimal stiffness at fixed rates.

### Position distribution in a generalised run and tumble process [article]

David S. Dean, Satya N. Majumdar, Hendrik Schawe
2020 arXiv   pre-print
For three special values n=1, n=2 and n=1/2 we compute the exact cumulant generating function of the position distribution at all times t.  ...  The tails of the position distribution at late times exhibit a large deviation form, p_n(x,t)∼[-γ t Φ_n(x/x^*(t))], where x^*(t)= v_0 t^n/Γ(n+1).  ...  /n 0 n ((1 − n)/g(n)) (1−n)/n (Γ(n + 1)) 1/n .  ...
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