PACS numbers: 05-40.-a; 05-60.-k
Keywords: ASEP, functional Bethe Ansatz, large deviations
I. INTRODUCTION
**The** one dimensional **asymmetric** simple **exclusion** **process** (ASEP) is one **of** **the** most simple examples **of** a classical interacting particles system exhibiting a non equilibrium steady state. This stochastic system has been studied much **in** **the** past, both **in** **the** mathematical [1, 2, 3] and physical literature [4, 5, 6, 7, 8, 9, 10, 11]. It consists **of** particles hopping locally on a one dimensional lattice, with an asymmetry between **the** forward and backward hopping rates. **The** **exclusion** constraint prevents **the** particles from moving to a site already occupied by another particle. **The** asymmetry between **the** hopping rates models **the** action **of** an external driving field **in** **the** bulk **of** **the** system, which maintains a permanent macroscopic **current** **in** **the** system. This **current** breaks **the** detailed balance and keeps **the** system out **of** equilibrium even **in** **the** stationary state. **The** special case for which **the** particles hop forward and backward with equal rates is called **the** symmetric simple **exclusion** **process** (SSEP). It corresponds to a situation for which **the** detailed balance holds **in** **the** bulk which means (**in** **the** absence **of** boundary conditions breaking **the** forward-backward symmetry) that **the** system reaches equilibrium **in** **the** long time limit. **In** this case, **the** system belongs to **the** universality class **of** **the** Edwards–Wilkinson (EW) equation [6, 12]. On **the** contrary, if **the** two hopping rates are different, detailed balance is broken and **the** system reaches **in** **the** long time limit a non equilibrium steady state characterized by **the** presence **of** a **current** **of** particles flowing through **the** system. **In** that case, **the** system belongs to **the** universality class **of** **the** Kardar–Parisi–Zhang (KPZ) equation [6, 13].

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independent **in** a system **of** size L, we shall therefore calculate F k only for
k ≤ L − 1. Thus, we need to consider reduced words **of** length j ≤ L − 1. Let W be such a word, and I(W ) be **the** set **of** indices **of** **the** operators M i
that compose W ; our aim is to find **the** expression **of** W and to calculate its prefactor from equation (15). Because **the** rules (2, 4) do not suppress or add any new index, **the** following property is true : if a word W ′ appearing

m(t, x) = x ∧ (1 − x) ∧ t , t > 0 , x ∈ [0, 1] , (1.7) see Figure 3. **The** macroscopic stationary state is reached at **the** finite time t f = 1/2.
One would naturally expect KPZ fluctuations to occur around this growing front. Let us recall a famous result **of** Bertini and Giacomin **in** that direction. They consider **the** WASEP on **the** infinite lattice Z, with upwards asymmetry ǫ. Starting from a flat initial profile, **the** hydrodynamic limit grows uniformly **in** space and at constant speed. Bertini and Giacomin look at fluctuations around this hydrodynamic limit according to **the** fol- lowing scaling: ǫ **in** height, ǫ −2 **in** space and ǫ −4 **in** time. They show that **the** rescaled interface converges to **the** Hopf-Cole solution **of** **the** KPZ equation on R. We stress that this height-space-time scaling is rigid **in** order to observe KPZ fluctuations **in** **the** exclu- sion **process**.

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scale behaviors are also observed **in** random matrix theory (e.g., **the** **asymmetric** tails **of** **the** Tracy-Widom distributions, c.f., [ TW94 ]), and **in** stochastic scalar conservation laws [ Mar10 ].
**The** existence **of** two-scale large deviations can be easily understood **in** **the** context **of** **the** **exclusion** processes. Recall that, for **the** TASEP, h(t, x) records **the** total number **of** particle passing through x. **The** lower deviation N 1 h(N, 0) < h(1, 0) − α (with α > 0) can be achieved by slowing down **the** Poisson clock at x = 0. Doing so creates a blockage and decelerates particle flow across x = 0. Such a situation involves slowdown **of** a single Poisson clock for time t ∈ [0, N ], and occurs with probability exp(−O(N )). On **the** other hand, for **the** upper deviation 1

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During **the** last two decades, substantial progress has been made towards a statistical theory **of** non-equilibrium systems [1–6]. Large deviation functions, that encode atypical fluctuations **of** a physical observable, are likely to be **the** best candidates to generalize **the** traditional thermodynamic potentials. Moreover, it has been proved that large deviations functions display symmetry prop- erties, called ‘Fluctuation Theorems’, that remain valid far from equilibrium [2]. These remarkable relations im- ply linear response theory **in** **the** vicinity **of** equilibrium. Hence, **the** determination **of** large deviations **in** a non- equilibrium system, whether theoretically, numerically, or experimentally, is a question **of** fundamental impor- tance [7–14].

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When α = 1/2, a sharp estimate on τ can be obtained directly from spectral considerations (Section 5.1), but when α ∈ [0, 1/2) we need a refinement **of** **the** strategy used **in** [LL18]: **The** first step (Proposition 10) is to obtain a control on **the** position **of** **the** leftmost particle which matches **the** lower bound provided by **the** hydrodynamic limit. This requires a new proof since **the** argument used **in** [LL18] is not sharp enough to cover all biases. **The** second step is to use contractive functions once **the** system is at macroscopic equilibrium, this is sufficient to treat most cases. A third and new step is required to treat **the** case when **the** bias b N **of** order log N/N or smaller: as we are working under **the** assumption (10) we only need to treat this case when k = N o(1) . **In** this third step we use diffusive estimates to control **the** hitting time **of** zero for **the** function f N,k (0) (h ∧ t ) − f N,k (0) (h ∨ t ) where f (0) was introduced **in** Subsection 3.4.

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o .
**In** particular, R (w n ) 2 dν ρ ≤ C(q, ρ)n 2α−2 and for 0 < α < 1, R (w n ) 2 dν ρ → 0
as n → ∞, and **the** lemma is trivial (there is no need **of** time integration). When 1 ≤ α < 2 an extra argument is needed. **The** idea is **the** following. For sites x near **the** origin, jumps **of** opposite sign cancel each other, as **in** **the** case **of** a diffusive system. Since **the** transition rate p(·) has infinite second moment, these cancellations do not hold for large x. Introduce a truncation around 0 **of** size n γ .

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Why don’t ALL committees reviewing protocols for funding reject those that cannot be applied to real people **in** **the** real world? **Of** course, “internal validity” is necessary, and while these may reduce type 1 and type 2 errors (false positives and false negatives), research is also open to what might be called type 3 errors. **In** these situations, we get **the** “right” answer but it’s to **the** wrong question – unlike work that leads to type 4 problems: **the** wrong answer to **the** wrong question.

Generalization **The** expression **of** **the** cumulant-generating function **of** multiinformation **in** **the** case **of** multivariate normal distributions is quite simple. This simplicity is mostly a consequence **of** **the** stability **of** **the** mul- tivariate normal family to most operations performed here: (i) **the** product **of** **the** marginals is also a multivariate normal distribution (this is strongly related to **the** fact that independence and uncorrelatedness are equivalent for multivariate normal distributions); (ii) **the** ratio **of** **the** joint distribu- tion to its marginals takes a simple form that is again closely related to **the** multivariate normal family; and (iii) **the** exponentiation **of** t times **the** log **of** this ratio still has **the** form **of** a multivariate normal distribution. It would be **of** interest to determine such a simplicity would still hold **in** more general settings such as more general distribution families and more general functions **of** **the** ratio f X (X)/ Q f X i (X i ).

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is purely based on Meissner expulsion.
**The** particular choice **of** orientation **of** **the** Pb stripes on top **of** **the** Al film has been made merely as a way to illustrate **the** proof **of** principle **of** our device. **The** observed, **current**- direction-dependent, vortex trapping could also be used to rectify vortex motion. 12,15,16 **In** addition, more controllable systems can be obtained by combining two independent **current** sources along **the** x and y directions. 17–19 **In** this case a vortex transistor can be developed where large voltage drops due to vortex motion along **the** y direction can be controlled with small forces or gating **in** **the** x direction. We expect **the** observed, **current**-direction-dependent, vortex-trapping mechanism can be magnified by making more repulsive cages with higher T c superconductors or magnetic templates with

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